# Complete Coloring Audits, Nonoptimal Upper Bounds and Color-Name Invariance

This batch has passed Schema, deduplication, source-snapshot, and independent-answer recomputation checks; it is still not marked as human-reviewed.

This question bank has corresponding bilingual step-by-step derivations. Source candidate and English ai_candidate states are retained. AI review, finite computation and Lean formal proof are labelled separately.

batch-20261014-advance-20261031-r05-graph-coloring-foundations · 2026-10-14 · 100 questions

This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.



## 1. Proper k-Coloring Means Edge Properness and At Most k Used Labels

Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5]]. Normalize each edge increasingly, then sort edges lexicographically; reversed duplicates and loops are forbidden. Keep isolated vertices in the carrier. No directed-graph conclusion is inferred. Total vertex colors=[0, 1, 0, 1, 0, 1] are labels, not cardinalities. Only adjacent endpoints must differ; nonadjacent vertices may share a color. A proper k-coloring uses at most k distinct labels, even if their numerical names exceed k−1. One such coloring gives only a chromatic-number upper bound, not optimality or a general greedy guarantee. k=2. Give (increasing_used_colors,normalized_conflicting_edges,edge_proper?,at_most_k?,proper_k_coloring?,proper_k_coloring_uses_at_most_k_colors).

View answer and explanation([0, 1], [], Yes, Yes, Yes, proper_k_coloring_uses_at_most_k_colors)

Audit every normalized edge and count distinct used labels separately. Both conditions are required; one passing coloring is not a proof of minimum chromatic number. The exact result, retaining the requested order, is ([0, 1], [], Yes, Yes, Yes, proper_k_coloring_uses_at_most_k_colors).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r05-graph-coloring-0001/)

Graph coloring · qf-mod31-coloring-proper-audit · candidate





## 2. Proper k-Coloring Means Edge Properness and At Most k Used Labels

Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5]]. Normalize each edge increasingly, then sort edges lexicographically; reversed duplicates and loops are forbidden. Keep isolated vertices in the carrier. No directed-graph conclusion is inferred. Total vertex colors=[1, 2, 1, 2, 1, 2] are labels, not cardinalities. Only adjacent endpoints must differ; nonadjacent vertices may share a color. A proper k-coloring uses at most k distinct labels, even if their numerical names exceed k−1. One such coloring gives only a chromatic-number upper bound, not optimality or a general greedy guarantee. k=3. Give (increasing_used_colors,normalized_conflicting_edges,edge_proper?,at_most_k?,proper_k_coloring?,proper_k_coloring_uses_at_most_k_colors).

View answer and explanation([1, 2], [], Yes, Yes, Yes, proper_k_coloring_uses_at_most_k_colors)

Audit every normalized edge and count distinct used labels separately. Both conditions are required; one passing coloring is not a proof of minimum chromatic number. The exact result, retaining the requested order, is ([1, 2], [], Yes, Yes, Yes, proper_k_coloring_uses_at_most_k_colors).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r05-graph-coloring-0002/)

Graph coloring · qf-mod31-coloring-proper-audit · candidate





## 3. Proper k-Coloring Means Edge Properness and At Most k Used Labels

Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5]]. Normalize each edge increasingly, then sort edges lexicographically; reversed duplicates and loops are forbidden. Keep isolated vertices in the carrier. No directed-graph conclusion is inferred. Total vertex colors=[2, 3, 2, 3, 2, 3] are labels, not cardinalities. Only adjacent endpoints must differ; nonadjacent vertices may share a color. A proper k-coloring uses at most k distinct labels, even if their numerical names exceed k−1. One such coloring gives only a chromatic-number upper bound, not optimality or a general greedy guarantee. k=4. Give (increasing_used_colors,normalized_conflicting_edges,edge_proper?,at_most_k?,proper_k_coloring?,proper_k_coloring_uses_at_most_k_colors).

View answer and explanation([2, 3], [], Yes, Yes, Yes, proper_k_coloring_uses_at_most_k_colors)

Audit every normalized edge and count distinct used labels separately. Both conditions are required; one passing coloring is not a proof of minimum chromatic number. The exact result, retaining the requested order, is ([2, 3], [], Yes, Yes, Yes, proper_k_coloring_uses_at_most_k_colors).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r05-graph-coloring-0003/)

Graph coloring · qf-mod31-coloring-proper-audit · candidate





## 4. Proper k-Coloring Means Edge Properness and At Most k Used Labels

Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5]]. Normalize each edge increasingly, then sort edges lexicographically; reversed duplicates and loops are forbidden. Keep isolated vertices in the carrier. No directed-graph conclusion is inferred. Total vertex colors=[3, 4, 3, 4, 3, 4] are labels, not cardinalities. Only adjacent endpoints must differ; nonadjacent vertices may share a color. A proper k-coloring uses at most k distinct labels, even if their numerical names exceed k−1. One such coloring gives only a chromatic-number upper bound, not optimality or a general greedy guarantee. k=2. Give (increasing_used_colors,normalized_conflicting_edges,edge_proper?,at_most_k?,proper_k_coloring?,proper_k_coloring_uses_at_most_k_colors).

View answer and explanation([3, 4], [], Yes, Yes, Yes, proper_k_coloring_uses_at_most_k_colors)

Audit every normalized edge and count distinct used labels separately. Both conditions are required; one passing coloring is not a proof of minimum chromatic number. The exact result, retaining the requested order, is ([3, 4], [], Yes, Yes, Yes, proper_k_coloring_uses_at_most_k_colors).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r05-graph-coloring-0004/)

Graph coloring · qf-mod31-coloring-proper-audit · candidate





## 5. Proper k-Coloring Means Edge Properness and At Most k Used Labels

Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5]]. Normalize each edge increasingly, then sort edges lexicographically; reversed duplicates and loops are forbidden. Keep isolated vertices in the carrier. No directed-graph conclusion is inferred. Total vertex colors=[4, 5, 4, 5, 4, 5] are labels, not cardinalities. Only adjacent endpoints must differ; nonadjacent vertices may share a color. A proper k-coloring uses at most k distinct labels, even if their numerical names exceed k−1. One such coloring gives only a chromatic-number upper bound, not optimality or a general greedy guarantee. k=3. Give (increasing_used_colors,normalized_conflicting_edges,edge_proper?,at_most_k?,proper_k_coloring?,proper_k_coloring_uses_at_most_k_colors).

View answer and explanation([4, 5], [], Yes, Yes, Yes, proper_k_coloring_uses_at_most_k_colors)

Audit every normalized edge and count distinct used labels separately. Both conditions are required; one passing coloring is not a proof of minimum chromatic number. The exact result, retaining the requested order, is ([4, 5], [], Yes, Yes, Yes, proper_k_coloring_uses_at_most_k_colors).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r05-graph-coloring-0005/)

Graph coloring · qf-mod31-coloring-proper-audit · candidate





## 6. Proper k-Coloring Means Edge Properness and At Most k Used Labels

Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5]]. Normalize each edge increasingly, then sort edges lexicographically; reversed duplicates and loops are forbidden. Keep isolated vertices in the carrier. No directed-graph conclusion is inferred. Total vertex colors=[5, 6, 5, 6, 5, 6] are labels, not cardinalities. Only adjacent endpoints must differ; nonadjacent vertices may share a color. A proper k-coloring uses at most k distinct labels, even if their numerical names exceed k−1. One such coloring gives only a chromatic-number upper bound, not optimality or a general greedy guarantee. k=4. Give (increasing_used_colors,normalized_conflicting_edges,edge_proper?,at_most_k?,proper_k_coloring?,proper_k_coloring_uses_at_most_k_colors).

View answer and explanation([5, 6], [], Yes, Yes, Yes, proper_k_coloring_uses_at_most_k_colors)

Audit every normalized edge and count distinct used labels separately. Both conditions are required; one passing coloring is not a proof of minimum chromatic number. The exact result, retaining the requested order, is ([5, 6], [], Yes, Yes, Yes, proper_k_coloring_uses_at_most_k_colors).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r05-graph-coloring-0006/)

Graph coloring · qf-mod31-coloring-proper-audit · candidate





## 7. Proper k-Coloring Means Edge Properness and At Most k Used Labels

Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5]]. Normalize each edge increasingly, then sort edges lexicographically; reversed duplicates and loops are forbidden. Keep isolated vertices in the carrier. No directed-graph conclusion is inferred. Total vertex colors=[6, 7, 6, 7, 6, 7] are labels, not cardinalities. Only adjacent endpoints must differ; nonadjacent vertices may share a color. A proper k-coloring uses at most k distinct labels, even if their numerical names exceed k−1. One such coloring gives only a chromatic-number upper bound, not optimality or a general greedy guarantee. k=2. Give (increasing_used_colors,normalized_conflicting_edges,edge_proper?,at_most_k?,proper_k_coloring?,proper_k_coloring_uses_at_most_k_colors).

View answer and explanation([6, 7], [], Yes, Yes, Yes, proper_k_coloring_uses_at_most_k_colors)

Audit every normalized edge and count distinct used labels separately. Both conditions are required; one passing coloring is not a proof of minimum chromatic number. The exact result, retaining the requested order, is ([6, 7], [], Yes, Yes, Yes, proper_k_coloring_uses_at_most_k_colors).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r05-graph-coloring-0007/)

Graph coloring · qf-mod31-coloring-proper-audit · candidate





## 8. Proper k-Coloring Means Edge Properness and At Most k Used Labels

Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5]]. Normalize each edge increasingly, then sort edges lexicographically; reversed duplicates and loops are forbidden. Keep isolated vertices in the carrier. No directed-graph conclusion is inferred. Total vertex colors=[7, 8, 7, 8, 7, 8] are labels, not cardinalities. Only adjacent endpoints must differ; nonadjacent vertices may share a color. A proper k-coloring uses at most k distinct labels, even if their numerical names exceed k−1. One such coloring gives only a chromatic-number upper bound, not optimality or a general greedy guarantee. k=3. Give (increasing_used_colors,normalized_conflicting_edges,edge_proper?,at_most_k?,proper_k_coloring?,proper_k_coloring_uses_at_most_k_colors).

View answer and explanation([7, 8], [], Yes, Yes, Yes, proper_k_coloring_uses_at_most_k_colors)

Audit every normalized edge and count distinct used labels separately. Both conditions are required; one passing coloring is not a proof of minimum chromatic number. The exact result, retaining the requested order, is ([7, 8], [], Yes, Yes, Yes, proper_k_coloring_uses_at_most_k_colors).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r05-graph-coloring-0008/)

Graph coloring · qf-mod31-coloring-proper-audit · candidate





## 9. Proper k-Coloring Means Edge Properness and At Most k Used Labels

Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5]]. Normalize each edge increasingly, then sort edges lexicographically; reversed duplicates and loops are forbidden. Keep isolated vertices in the carrier. No directed-graph conclusion is inferred. Total vertex colors=[8, 9, 8, 9, 8, 9] are labels, not cardinalities. Only adjacent endpoints must differ; nonadjacent vertices may share a color. A proper k-coloring uses at most k distinct labels, even if their numerical names exceed k−1. One such coloring gives only a chromatic-number upper bound, not optimality or a general greedy guarantee. k=4. Give (increasing_used_colors,normalized_conflicting_edges,edge_proper?,at_most_k?,proper_k_coloring?,proper_k_coloring_uses_at_most_k_colors).

View answer and explanation([8, 9], [], Yes, Yes, Yes, proper_k_coloring_uses_at_most_k_colors)

Audit every normalized edge and count distinct used labels separately. Both conditions are required; one passing coloring is not a proof of minimum chromatic number. The exact result, retaining the requested order, is ([8, 9], [], Yes, Yes, Yes, proper_k_coloring_uses_at_most_k_colors).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r05-graph-coloring-0009/)

Graph coloring · qf-mod31-coloring-proper-audit · candidate





## 10. Proper k-Coloring Means Edge Properness and At Most k Used Labels

Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5]]. Normalize each edge increasingly, then sort edges lexicographically; reversed duplicates and loops are forbidden. Keep isolated vertices in the carrier. No directed-graph conclusion is inferred. Total vertex colors=[9, 10, 9, 10, 9, 10] are labels, not cardinalities. Only adjacent endpoints must differ; nonadjacent vertices may share a color. A proper k-coloring uses at most k distinct labels, even if their numerical names exceed k−1. One such coloring gives only a chromatic-number upper bound, not optimality or a general greedy guarantee. k=2. Give (increasing_used_colors,normalized_conflicting_edges,edge_proper?,at_most_k?,proper_k_coloring?,proper_k_coloring_uses_at_most_k_colors).

View answer and explanation([9, 10], [], Yes, Yes, Yes, proper_k_coloring_uses_at_most_k_colors)

Audit every normalized edge and count distinct used labels separately. Both conditions are required; one passing coloring is not a proof of minimum chromatic number. The exact result, retaining the requested order, is ([9, 10], [], Yes, Yes, Yes, proper_k_coloring_uses_at_most_k_colors).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r05-graph-coloring-0010/)

Graph coloring · qf-mod31-coloring-proper-audit · candidate





## 11. Proper k-Coloring Means Edge Properness and At Most k Used Labels

Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5]]. Normalize each edge increasingly, then sort edges lexicographically; reversed duplicates and loops are forbidden. Keep isolated vertices in the carrier. No directed-graph conclusion is inferred. Total vertex colors=[10, 11, 10, 11, 10, 11] are labels, not cardinalities. Only adjacent endpoints must differ; nonadjacent vertices may share a color. A proper k-coloring uses at most k distinct labels, even if their numerical names exceed k−1. One such coloring gives only a chromatic-number upper bound, not optimality or a general greedy guarantee. k=3. Give (increasing_used_colors,normalized_conflicting_edges,edge_proper?,at_most_k?,proper_k_coloring?,proper_k_coloring_uses_at_most_k_colors).

View answer and explanation([10, 11], [], Yes, Yes, Yes, proper_k_coloring_uses_at_most_k_colors)

Audit every normalized edge and count distinct used labels separately. Both conditions are required; one passing coloring is not a proof of minimum chromatic number. The exact result, retaining the requested order, is ([10, 11], [], Yes, Yes, Yes, proper_k_coloring_uses_at_most_k_colors).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r05-graph-coloring-0011/)

Graph coloring · qf-mod31-coloring-proper-audit · candidate





## 12. Proper k-Coloring Means Edge Properness and At Most k Used Labels

Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5]]. Normalize each edge increasingly, then sort edges lexicographically; reversed duplicates and loops are forbidden. Keep isolated vertices in the carrier. No directed-graph conclusion is inferred. Total vertex colors=[11, 12, 11, 12, 11, 12] are labels, not cardinalities. Only adjacent endpoints must differ; nonadjacent vertices may share a color. A proper k-coloring uses at most k distinct labels, even if their numerical names exceed k−1. One such coloring gives only a chromatic-number upper bound, not optimality or a general greedy guarantee. k=4. Give (increasing_used_colors,normalized_conflicting_edges,edge_proper?,at_most_k?,proper_k_coloring?,proper_k_coloring_uses_at_most_k_colors).

View answer and explanation([11, 12], [], Yes, Yes, Yes, proper_k_coloring_uses_at_most_k_colors)

Audit every normalized edge and count distinct used labels separately. Both conditions are required; one passing coloring is not a proof of minimum chromatic number. The exact result, retaining the requested order, is ([11, 12], [], Yes, Yes, Yes, proper_k_coloring_uses_at_most_k_colors).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r05-graph-coloring-0012/)

Graph coloring · qf-mod31-coloring-proper-audit · candidate





## 13. Proper k-Coloring Means Edge Properness and At Most k Used Labels

Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5]]. Normalize each edge increasingly, then sort edges lexicographically; reversed duplicates and loops are forbidden. Keep isolated vertices in the carrier. No directed-graph conclusion is inferred. Total vertex colors=[12, 13, 12, 13, 12, 13] are labels, not cardinalities. Only adjacent endpoints must differ; nonadjacent vertices may share a color. A proper k-coloring uses at most k distinct labels, even if their numerical names exceed k−1. One such coloring gives only a chromatic-number upper bound, not optimality or a general greedy guarantee. k=2. Give (increasing_used_colors,normalized_conflicting_edges,edge_proper?,at_most_k?,proper_k_coloring?,proper_k_coloring_uses_at_most_k_colors).

View answer and explanation([12, 13], [], Yes, Yes, Yes, proper_k_coloring_uses_at_most_k_colors)

Audit every normalized edge and count distinct used labels separately. Both conditions are required; one passing coloring is not a proof of minimum chromatic number. The exact result, retaining the requested order, is ([12, 13], [], Yes, Yes, Yes, proper_k_coloring_uses_at_most_k_colors).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r05-graph-coloring-0013/)

Graph coloring · qf-mod31-coloring-proper-audit · candidate





## 14. Proper k-Coloring Means Edge Properness and At Most k Used Labels

Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5]]. Normalize each edge increasingly, then sort edges lexicographically; reversed duplicates and loops are forbidden. Keep isolated vertices in the carrier. No directed-graph conclusion is inferred. Total vertex colors=[13, 14, 13, 14, 13, 14] are labels, not cardinalities. Only adjacent endpoints must differ; nonadjacent vertices may share a color. A proper k-coloring uses at most k distinct labels, even if their numerical names exceed k−1. One such coloring gives only a chromatic-number upper bound, not optimality or a general greedy guarantee. k=3. Give (increasing_used_colors,normalized_conflicting_edges,edge_proper?,at_most_k?,proper_k_coloring?,proper_k_coloring_uses_at_most_k_colors).

View answer and explanation([13, 14], [], Yes, Yes, Yes, proper_k_coloring_uses_at_most_k_colors)

Audit every normalized edge and count distinct used labels separately. Both conditions are required; one passing coloring is not a proof of minimum chromatic number. The exact result, retaining the requested order, is ([13, 14], [], Yes, Yes, Yes, proper_k_coloring_uses_at_most_k_colors).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r05-graph-coloring-0014/)

Graph coloring · qf-mod31-coloring-proper-audit · candidate





## 15. Proper k-Coloring Means Edge Properness and At Most k Used Labels

Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5]]. Normalize each edge increasingly, then sort edges lexicographically; reversed duplicates and loops are forbidden. Keep isolated vertices in the carrier. No directed-graph conclusion is inferred. Total vertex colors=[14, 15, 14, 15, 14, 15] are labels, not cardinalities. Only adjacent endpoints must differ; nonadjacent vertices may share a color. A proper k-coloring uses at most k distinct labels, even if their numerical names exceed k−1. One such coloring gives only a chromatic-number upper bound, not optimality or a general greedy guarantee. k=4. Give (increasing_used_colors,normalized_conflicting_edges,edge_proper?,at_most_k?,proper_k_coloring?,proper_k_coloring_uses_at_most_k_colors).

View answer and explanation([14, 15], [], Yes, Yes, Yes, proper_k_coloring_uses_at_most_k_colors)

Audit every normalized edge and count distinct used labels separately. Both conditions are required; one passing coloring is not a proof of minimum chromatic number. The exact result, retaining the requested order, is ([14, 15], [], Yes, Yes, Yes, proper_k_coloring_uses_at_most_k_colors).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r05-graph-coloring-0015/)

Graph coloring · qf-mod31-coloring-proper-audit · candidate





## 16. Proper k-Coloring Means Edge Properness and At Most k Used Labels

Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5]]. Normalize each edge increasingly, then sort edges lexicographically; reversed duplicates and loops are forbidden. Keep isolated vertices in the carrier. No directed-graph conclusion is inferred. Total vertex colors=[15, 16, 15, 16, 15, 16] are labels, not cardinalities. Only adjacent endpoints must differ; nonadjacent vertices may share a color. A proper k-coloring uses at most k distinct labels, even if their numerical names exceed k−1. One such coloring gives only a chromatic-number upper bound, not optimality or a general greedy guarantee. k=2. Give (increasing_used_colors,normalized_conflicting_edges,edge_proper?,at_most_k?,proper_k_coloring?,proper_k_coloring_uses_at_most_k_colors).

View answer and explanation([15, 16], [], Yes, Yes, Yes, proper_k_coloring_uses_at_most_k_colors)

Audit every normalized edge and count distinct used labels separately. Both conditions are required; one passing coloring is not a proof of minimum chromatic number. The exact result, retaining the requested order, is ([15, 16], [], Yes, Yes, Yes, proper_k_coloring_uses_at_most_k_colors).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r05-graph-coloring-0016/)

Graph coloring · qf-mod31-coloring-proper-audit · candidate





## 17. Proper k-Coloring Means Edge Properness and At Most k Used Labels

Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5]]. Normalize each edge increasingly, then sort edges lexicographically; reversed duplicates and loops are forbidden. Keep isolated vertices in the carrier. No directed-graph conclusion is inferred. Total vertex colors=[16, 17, 16, 17, 16, 17] are labels, not cardinalities. Only adjacent endpoints must differ; nonadjacent vertices may share a color. A proper k-coloring uses at most k distinct labels, even if their numerical names exceed k−1. One such coloring gives only a chromatic-number upper bound, not optimality or a general greedy guarantee. k=3. Give (increasing_used_colors,normalized_conflicting_edges,edge_proper?,at_most_k?,proper_k_coloring?,proper_k_coloring_uses_at_most_k_colors).

View answer and explanation([16, 17], [], Yes, Yes, Yes, proper_k_coloring_uses_at_most_k_colors)

Audit every normalized edge and count distinct used labels separately. Both conditions are required; one passing coloring is not a proof of minimum chromatic number. The exact result, retaining the requested order, is ([16, 17], [], Yes, Yes, Yes, proper_k_coloring_uses_at_most_k_colors).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r05-graph-coloring-0017/)

Graph coloring · qf-mod31-coloring-proper-audit · candidate





## 18. Proper k-Coloring Means Edge Properness and At Most k Used Labels

Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5]]. Normalize each edge increasingly, then sort edges lexicographically; reversed duplicates and loops are forbidden. Keep isolated vertices in the carrier. No directed-graph conclusion is inferred. Total vertex colors=[17, 18, 17, 18, 17, 18] are labels, not cardinalities. Only adjacent endpoints must differ; nonadjacent vertices may share a color. A proper k-coloring uses at most k distinct labels, even if their numerical names exceed k−1. One such coloring gives only a chromatic-number upper bound, not optimality or a general greedy guarantee. k=4. Give (increasing_used_colors,normalized_conflicting_edges,edge_proper?,at_most_k?,proper_k_coloring?,proper_k_coloring_uses_at_most_k_colors).

View answer and explanation([17, 18], [], Yes, Yes, Yes, proper_k_coloring_uses_at_most_k_colors)

Audit every normalized edge and count distinct used labels separately. Both conditions are required; one passing coloring is not a proof of minimum chromatic number. The exact result, retaining the requested order, is ([17, 18], [], Yes, Yes, Yes, proper_k_coloring_uses_at_most_k_colors).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r05-graph-coloring-0018/)

Graph coloring · qf-mod31-coloring-proper-audit · candidate





## 19. Proper k-Coloring Means Edge Properness and At Most k Used Labels

Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5]]. Normalize each edge increasingly, then sort edges lexicographically; reversed duplicates and loops are forbidden. Keep isolated vertices in the carrier. No directed-graph conclusion is inferred. Total vertex colors=[18, 19, 18, 19, 18, 19] are labels, not cardinalities. Only adjacent endpoints must differ; nonadjacent vertices may share a color. A proper k-coloring uses at most k distinct labels, even if their numerical names exceed k−1. One such coloring gives only a chromatic-number upper bound, not optimality or a general greedy guarantee. k=2. Give (increasing_used_colors,normalized_conflicting_edges,edge_proper?,at_most_k?,proper_k_coloring?,proper_k_coloring_uses_at_most_k_colors).

View answer and explanation([18, 19], [], Yes, Yes, Yes, proper_k_coloring_uses_at_most_k_colors)

Audit every normalized edge and count distinct used labels separately. Both conditions are required; one passing coloring is not a proof of minimum chromatic number. The exact result, retaining the requested order, is ([18, 19], [], Yes, Yes, Yes, proper_k_coloring_uses_at_most_k_colors).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r05-graph-coloring-0019/)

Graph coloring · qf-mod31-coloring-proper-audit · candidate





## 20. Proper k-Coloring Means Edge Properness and At Most k Used Labels

Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5]]. Normalize each edge increasingly, then sort edges lexicographically; reversed duplicates and loops are forbidden. Keep isolated vertices in the carrier. No directed-graph conclusion is inferred. Total vertex colors=[19, 20, 19, 20, 19, 20] are labels, not cardinalities. Only adjacent endpoints must differ; nonadjacent vertices may share a color. A proper k-coloring uses at most k distinct labels, even if their numerical names exceed k−1. One such coloring gives only a chromatic-number upper bound, not optimality or a general greedy guarantee. k=3. Give (increasing_used_colors,normalized_conflicting_edges,edge_proper?,at_most_k?,proper_k_coloring?,proper_k_coloring_uses_at_most_k_colors).

View answer and explanation([19, 20], [], Yes, Yes, Yes, proper_k_coloring_uses_at_most_k_colors)

Audit every normalized edge and count distinct used labels separately. Both conditions are required; one passing coloring is not a proof of minimum chromatic number. The exact result, retaining the requested order, is ([19, 20], [], Yes, Yes, Yes, proper_k_coloring_uses_at_most_k_colors).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r05-graph-coloring-0020/)

Graph coloring · qf-mod31-coloring-proper-audit · candidate





## 21. Proper k-Coloring Means Edge Properness and At Most k Used Labels

Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5]]. Normalize each edge increasingly, then sort edges lexicographically; reversed duplicates and loops are forbidden. Keep isolated vertices in the carrier. No directed-graph conclusion is inferred. Total vertex colors=[20, 21, 20, 21, 20, 21] are labels, not cardinalities. Only adjacent endpoints must differ; nonadjacent vertices may share a color. A proper k-coloring uses at most k distinct labels, even if their numerical names exceed k−1. One such coloring gives only a chromatic-number upper bound, not optimality or a general greedy guarantee. k=4. Give (increasing_used_colors,normalized_conflicting_edges,edge_proper?,at_most_k?,proper_k_coloring?,proper_k_coloring_uses_at_most_k_colors).

View answer and explanation([20, 21], [], Yes, Yes, Yes, proper_k_coloring_uses_at_most_k_colors)

Audit every normalized edge and count distinct used labels separately. Both conditions are required; one passing coloring is not a proof of minimum chromatic number. The exact result, retaining the requested order, is ([20, 21], [], Yes, Yes, Yes, proper_k_coloring_uses_at_most_k_colors).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r05-graph-coloring-0021/)

Graph coloring · qf-mod31-coloring-proper-audit · candidate





## 22. Proper k-Coloring Means Edge Properness and At Most k Used Labels

Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5]]. Normalize each edge increasingly, then sort edges lexicographically; reversed duplicates and loops are forbidden. Keep isolated vertices in the carrier. No directed-graph conclusion is inferred. Total vertex colors=[21, 22, 21, 22, 21, 22] are labels, not cardinalities. Only adjacent endpoints must differ; nonadjacent vertices may share a color. A proper k-coloring uses at most k distinct labels, even if their numerical names exceed k−1. One such coloring gives only a chromatic-number upper bound, not optimality or a general greedy guarantee. k=2. Give (increasing_used_colors,normalized_conflicting_edges,edge_proper?,at_most_k?,proper_k_coloring?,proper_k_coloring_uses_at_most_k_colors).

View answer and explanation([21, 22], [], Yes, Yes, Yes, proper_k_coloring_uses_at_most_k_colors)

Audit every normalized edge and count distinct used labels separately. Both conditions are required; one passing coloring is not a proof of minimum chromatic number. The exact result, retaining the requested order, is ([21, 22], [], Yes, Yes, Yes, proper_k_coloring_uses_at_most_k_colors).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r05-graph-coloring-0022/)

Graph coloring · qf-mod31-coloring-proper-audit · candidate





## 23. Proper k-Coloring Means Edge Properness and At Most k Used Labels

Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5]]. Normalize each edge increasingly, then sort edges lexicographically; reversed duplicates and loops are forbidden. Keep isolated vertices in the carrier. No directed-graph conclusion is inferred. Total vertex colors=[22, 23, 22, 23, 22, 23] are labels, not cardinalities. Only adjacent endpoints must differ; nonadjacent vertices may share a color. A proper k-coloring uses at most k distinct labels, even if their numerical names exceed k−1. One such coloring gives only a chromatic-number upper bound, not optimality or a general greedy guarantee. k=3. Give (increasing_used_colors,normalized_conflicting_edges,edge_proper?,at_most_k?,proper_k_coloring?,proper_k_coloring_uses_at_most_k_colors).

View answer and explanation([22, 23], [], Yes, Yes, Yes, proper_k_coloring_uses_at_most_k_colors)

Audit every normalized edge and count distinct used labels separately. Both conditions are required; one passing coloring is not a proof of minimum chromatic number. The exact result, retaining the requested order, is ([22, 23], [], Yes, Yes, Yes, proper_k_coloring_uses_at_most_k_colors).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r05-graph-coloring-0023/)

Graph coloring · qf-mod31-coloring-proper-audit · candidate





## 24. Proper k-Coloring Means Edge Properness and At Most k Used Labels

Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5]]. Normalize each edge increasingly, then sort edges lexicographically; reversed duplicates and loops are forbidden. Keep isolated vertices in the carrier. No directed-graph conclusion is inferred. Total vertex colors=[23, 24, 23, 24, 23, 24] are labels, not cardinalities. Only adjacent endpoints must differ; nonadjacent vertices may share a color. A proper k-coloring uses at most k distinct labels, even if their numerical names exceed k−1. One such coloring gives only a chromatic-number upper bound, not optimality or a general greedy guarantee. k=4. Give (increasing_used_colors,normalized_conflicting_edges,edge_proper?,at_most_k?,proper_k_coloring?,proper_k_coloring_uses_at_most_k_colors).

View answer and explanation([23, 24], [], Yes, Yes, Yes, proper_k_coloring_uses_at_most_k_colors)

Audit every normalized edge and count distinct used labels separately. Both conditions are required; one passing coloring is not a proof of minimum chromatic number. The exact result, retaining the requested order, is ([23, 24], [], Yes, Yes, Yes, proper_k_coloring_uses_at_most_k_colors).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r05-graph-coloring-0024/)

Graph coloring · qf-mod31-coloring-proper-audit · candidate





## 25. Proper k-Coloring Means Edge Properness and At Most k Used Labels

Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5]]. Normalize each edge increasingly, then sort edges lexicographically; reversed duplicates and loops are forbidden. Keep isolated vertices in the carrier. No directed-graph conclusion is inferred. Total vertex colors=[24, 25, 24, 25, 24, 25] are labels, not cardinalities. Only adjacent endpoints must differ; nonadjacent vertices may share a color. A proper k-coloring uses at most k distinct labels, even if their numerical names exceed k−1. One such coloring gives only a chromatic-number upper bound, not optimality or a general greedy guarantee. k=2. Give (increasing_used_colors,normalized_conflicting_edges,edge_proper?,at_most_k?,proper_k_coloring?,proper_k_coloring_uses_at_most_k_colors).

View answer and explanation([24, 25], [], Yes, Yes, Yes, proper_k_coloring_uses_at_most_k_colors)

Audit every normalized edge and count distinct used labels separately. Both conditions are required; one passing coloring is not a proof of minimum chromatic number. The exact result, retaining the requested order, is ([24, 25], [], Yes, Yes, Yes, proper_k_coloring_uses_at_most_k_colors).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r05-graph-coloring-0025/)

Graph coloring · qf-mod31-coloring-proper-audit · candidate





## 26. List Every Monochromatic Edge, Not Every Repeated Color

Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5]]. Normalize each edge increasingly, then sort edges lexicographically; reversed duplicates and loops are forbidden. Keep isolated vertices in the carrier. No directed-graph conclusion is inferred. Total vertex colors=[0, 0, 1, 0, 1, 0] are labels, not cardinalities. Only adjacent endpoints must differ; nonadjacent vertices may share a color. A proper k-coloring uses at most k distinct labels, even if their numerical names exceed k−1. One such coloring gives only a chromatic-number upper bound, not optimality or a general greedy guarantee. Give (conflicting_edges,count,proper?), with edges in normalized sorted order.

View answer and explanation([[0, 1]], 1, No)

Only same-colored endpoints joined by an edge are conflicts. Reusing a color on nonadjacent vertices is allowed. The exact result, retaining the requested order, is ([[0, 1]], 1, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r05-graph-coloring-0026/)

Graph coloring · qf-mod31-coloring-conflict · candidate





## 27. List Every Monochromatic Edge, Not Every Repeated Color

Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5]]. Normalize each edge increasingly, then sort edges lexicographically; reversed duplicates and loops are forbidden. Keep isolated vertices in the carrier. No directed-graph conclusion is inferred. Total vertex colors=[1, 1, 2, 1, 2, 1] are labels, not cardinalities. Only adjacent endpoints must differ; nonadjacent vertices may share a color. A proper k-coloring uses at most k distinct labels, even if their numerical names exceed k−1. One such coloring gives only a chromatic-number upper bound, not optimality or a general greedy guarantee. Give (conflicting_edges,count,proper?), with edges in normalized sorted order.

View answer and explanation([[0, 1]], 1, No)

Only same-colored endpoints joined by an edge are conflicts. Reusing a color on nonadjacent vertices is allowed. The exact result, retaining the requested order, is ([[0, 1]], 1, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r05-graph-coloring-0027/)

Graph coloring · qf-mod31-coloring-conflict · candidate





## 28. List Every Monochromatic Edge, Not Every Repeated Color

Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5]]. Normalize each edge increasingly, then sort edges lexicographically; reversed duplicates and loops are forbidden. Keep isolated vertices in the carrier. No directed-graph conclusion is inferred. Total vertex colors=[2, 2, 3, 2, 3, 2] are labels, not cardinalities. Only adjacent endpoints must differ; nonadjacent vertices may share a color. A proper k-coloring uses at most k distinct labels, even if their numerical names exceed k−1. One such coloring gives only a chromatic-number upper bound, not optimality or a general greedy guarantee. Give (conflicting_edges,count,proper?), with edges in normalized sorted order.

View answer and explanation([[0, 1]], 1, No)

Only same-colored endpoints joined by an edge are conflicts. Reusing a color on nonadjacent vertices is allowed. The exact result, retaining the requested order, is ([[0, 1]], 1, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r05-graph-coloring-0028/)

Graph coloring · qf-mod31-coloring-conflict · candidate





## 29. List Every Monochromatic Edge, Not Every Repeated Color

Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5]]. Normalize each edge increasingly, then sort edges lexicographically; reversed duplicates and loops are forbidden. Keep isolated vertices in the carrier. No directed-graph conclusion is inferred. Total vertex colors=[3, 3, 4, 3, 4, 3] are labels, not cardinalities. Only adjacent endpoints must differ; nonadjacent vertices may share a color. A proper k-coloring uses at most k distinct labels, even if their numerical names exceed k−1. One such coloring gives only a chromatic-number upper bound, not optimality or a general greedy guarantee. Give (conflicting_edges,count,proper?), with edges in normalized sorted order.

View answer and explanation([[0, 1]], 1, No)

Only same-colored endpoints joined by an edge are conflicts. Reusing a color on nonadjacent vertices is allowed. The exact result, retaining the requested order, is ([[0, 1]], 1, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r05-graph-coloring-0029/)

Graph coloring · qf-mod31-coloring-conflict · candidate





## 30. List Every Monochromatic Edge, Not Every Repeated Color

Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5]]. Normalize each edge increasingly, then sort edges lexicographically; reversed duplicates and loops are forbidden. Keep isolated vertices in the carrier. No directed-graph conclusion is inferred. Total vertex colors=[4, 4, 5, 4, 5, 4] are labels, not cardinalities. Only adjacent endpoints must differ; nonadjacent vertices may share a color. A proper k-coloring uses at most k distinct labels, even if their numerical names exceed k−1. One such coloring gives only a chromatic-number upper bound, not optimality or a general greedy guarantee. Give (conflicting_edges,count,proper?), with edges in normalized sorted order.

View answer and explanation([[0, 1]], 1, No)

Only same-colored endpoints joined by an edge are conflicts. Reusing a color on nonadjacent vertices is allowed. The exact result, retaining the requested order, is ([[0, 1]], 1, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r05-graph-coloring-0030/)

Graph coloring · qf-mod31-coloring-conflict · candidate





## 31. List Every Monochromatic Edge, Not Every Repeated Color

Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5]]. Normalize each edge increasingly, then sort edges lexicographically; reversed duplicates and loops are forbidden. Keep isolated vertices in the carrier. No directed-graph conclusion is inferred. Total vertex colors=[5, 5, 6, 5, 6, 5] are labels, not cardinalities. Only adjacent endpoints must differ; nonadjacent vertices may share a color. A proper k-coloring uses at most k distinct labels, even if their numerical names exceed k−1. One such coloring gives only a chromatic-number upper bound, not optimality or a general greedy guarantee. Give (conflicting_edges,count,proper?), with edges in normalized sorted order.

View answer and explanation([[0, 1]], 1, No)

Only same-colored endpoints joined by an edge are conflicts. Reusing a color on nonadjacent vertices is allowed. The exact result, retaining the requested order, is ([[0, 1]], 1, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r05-graph-coloring-0031/)

Graph coloring · qf-mod31-coloring-conflict · candidate





## 32. List Every Monochromatic Edge, Not Every Repeated Color

Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5]]. Normalize each edge increasingly, then sort edges lexicographically; reversed duplicates and loops are forbidden. Keep isolated vertices in the carrier. No directed-graph conclusion is inferred. Total vertex colors=[6, 6, 7, 6, 7, 6] are labels, not cardinalities. Only adjacent endpoints must differ; nonadjacent vertices may share a color. A proper k-coloring uses at most k distinct labels, even if their numerical names exceed k−1. One such coloring gives only a chromatic-number upper bound, not optimality or a general greedy guarantee. Give (conflicting_edges,count,proper?), with edges in normalized sorted order.

View answer and explanation([[0, 1]], 1, No)

Only same-colored endpoints joined by an edge are conflicts. Reusing a color on nonadjacent vertices is allowed. The exact result, retaining the requested order, is ([[0, 1]], 1, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r05-graph-coloring-0032/)

Graph coloring · qf-mod31-coloring-conflict · candidate





## 33. List Every Monochromatic Edge, Not Every Repeated Color

Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5]]. Normalize each edge increasingly, then sort edges lexicographically; reversed duplicates and loops are forbidden. Keep isolated vertices in the carrier. No directed-graph conclusion is inferred. Total vertex colors=[7, 7, 8, 7, 8, 7] are labels, not cardinalities. Only adjacent endpoints must differ; nonadjacent vertices may share a color. A proper k-coloring uses at most k distinct labels, even if their numerical names exceed k−1. One such coloring gives only a chromatic-number upper bound, not optimality or a general greedy guarantee. Give (conflicting_edges,count,proper?), with edges in normalized sorted order.

View answer and explanation([[0, 1]], 1, No)

Only same-colored endpoints joined by an edge are conflicts. Reusing a color on nonadjacent vertices is allowed. The exact result, retaining the requested order, is ([[0, 1]], 1, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r05-graph-coloring-0033/)

Graph coloring · qf-mod31-coloring-conflict · candidate





## 34. List Every Monochromatic Edge, Not Every Repeated Color

Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5]]. Normalize each edge increasingly, then sort edges lexicographically; reversed duplicates and loops are forbidden. Keep isolated vertices in the carrier. No directed-graph conclusion is inferred. Total vertex colors=[8, 8, 9, 8, 9, 8] are labels, not cardinalities. Only adjacent endpoints must differ; nonadjacent vertices may share a color. A proper k-coloring uses at most k distinct labels, even if their numerical names exceed k−1. One such coloring gives only a chromatic-number upper bound, not optimality or a general greedy guarantee. Give (conflicting_edges,count,proper?), with edges in normalized sorted order.

View answer and explanation([[0, 1]], 1, No)

Only same-colored endpoints joined by an edge are conflicts. Reusing a color on nonadjacent vertices is allowed. The exact result, retaining the requested order, is ([[0, 1]], 1, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r05-graph-coloring-0034/)

Graph coloring · qf-mod31-coloring-conflict · candidate





## 35. List Every Monochromatic Edge, Not Every Repeated Color

Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5]]. Normalize each edge increasingly, then sort edges lexicographically; reversed duplicates and loops are forbidden. Keep isolated vertices in the carrier. No directed-graph conclusion is inferred. Total vertex colors=[9, 9, 10, 9, 10, 9] are labels, not cardinalities. Only adjacent endpoints must differ; nonadjacent vertices may share a color. A proper k-coloring uses at most k distinct labels, even if their numerical names exceed k−1. One such coloring gives only a chromatic-number upper bound, not optimality or a general greedy guarantee. Give (conflicting_edges,count,proper?), with edges in normalized sorted order.

View answer and explanation([[0, 1]], 1, No)

Only same-colored endpoints joined by an edge are conflicts. Reusing a color on nonadjacent vertices is allowed. The exact result, retaining the requested order, is ([[0, 1]], 1, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r05-graph-coloring-0035/)

Graph coloring · qf-mod31-coloring-conflict · candidate





## 36. List Every Monochromatic Edge, Not Every Repeated Color

Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5]]. Normalize each edge increasingly, then sort edges lexicographically; reversed duplicates and loops are forbidden. Keep isolated vertices in the carrier. No directed-graph conclusion is inferred. Total vertex colors=[10, 10, 11, 10, 11, 10] are labels, not cardinalities. Only adjacent endpoints must differ; nonadjacent vertices may share a color. A proper k-coloring uses at most k distinct labels, even if their numerical names exceed k−1. One such coloring gives only a chromatic-number upper bound, not optimality or a general greedy guarantee. Give (conflicting_edges,count,proper?), with edges in normalized sorted order.

View answer and explanation([[0, 1]], 1, No)

Only same-colored endpoints joined by an edge are conflicts. Reusing a color on nonadjacent vertices is allowed. The exact result, retaining the requested order, is ([[0, 1]], 1, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r05-graph-coloring-0036/)

Graph coloring · qf-mod31-coloring-conflict · candidate





## 37. List Every Monochromatic Edge, Not Every Repeated Color

Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5]]. Normalize each edge increasingly, then sort edges lexicographically; reversed duplicates and loops are forbidden. Keep isolated vertices in the carrier. No directed-graph conclusion is inferred. Total vertex colors=[11, 11, 12, 11, 12, 11] are labels, not cardinalities. Only adjacent endpoints must differ; nonadjacent vertices may share a color. A proper k-coloring uses at most k distinct labels, even if their numerical names exceed k−1. One such coloring gives only a chromatic-number upper bound, not optimality or a general greedy guarantee. Give (conflicting_edges,count,proper?), with edges in normalized sorted order.

View answer and explanation([[0, 1]], 1, No)

Only same-colored endpoints joined by an edge are conflicts. Reusing a color on nonadjacent vertices is allowed. The exact result, retaining the requested order, is ([[0, 1]], 1, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r05-graph-coloring-0037/)

Graph coloring · qf-mod31-coloring-conflict · candidate





## 38. List Every Monochromatic Edge, Not Every Repeated Color

Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5]]. Normalize each edge increasingly, then sort edges lexicographically; reversed duplicates and loops are forbidden. Keep isolated vertices in the carrier. No directed-graph conclusion is inferred. Total vertex colors=[12, 12, 13, 12, 13, 12] are labels, not cardinalities. Only adjacent endpoints must differ; nonadjacent vertices may share a color. A proper k-coloring uses at most k distinct labels, even if their numerical names exceed k−1. One such coloring gives only a chromatic-number upper bound, not optimality or a general greedy guarantee. Give (conflicting_edges,count,proper?), with edges in normalized sorted order.

View answer and explanation([[0, 1]], 1, No)

Only same-colored endpoints joined by an edge are conflicts. Reusing a color on nonadjacent vertices is allowed. The exact result, retaining the requested order, is ([[0, 1]], 1, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r05-graph-coloring-0038/)

Graph coloring · qf-mod31-coloring-conflict · candidate





## 39. List Every Monochromatic Edge, Not Every Repeated Color

Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5]]. Normalize each edge increasingly, then sort edges lexicographically; reversed duplicates and loops are forbidden. Keep isolated vertices in the carrier. No directed-graph conclusion is inferred. Total vertex colors=[13, 13, 14, 13, 14, 13] are labels, not cardinalities. Only adjacent endpoints must differ; nonadjacent vertices may share a color. A proper k-coloring uses at most k distinct labels, even if their numerical names exceed k−1. One such coloring gives only a chromatic-number upper bound, not optimality or a general greedy guarantee. Give (conflicting_edges,count,proper?), with edges in normalized sorted order.

View answer and explanation([[0, 1]], 1, No)

Only same-colored endpoints joined by an edge are conflicts. Reusing a color on nonadjacent vertices is allowed. The exact result, retaining the requested order, is ([[0, 1]], 1, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r05-graph-coloring-0039/)

Graph coloring · qf-mod31-coloring-conflict · candidate





## 40. List Every Monochromatic Edge, Not Every Repeated Color

Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5]]. Normalize each edge increasingly, then sort edges lexicographically; reversed duplicates and loops are forbidden. Keep isolated vertices in the carrier. No directed-graph conclusion is inferred. Total vertex colors=[14, 14, 15, 14, 15, 14] are labels, not cardinalities. Only adjacent endpoints must differ; nonadjacent vertices may share a color. A proper k-coloring uses at most k distinct labels, even if their numerical names exceed k−1. One such coloring gives only a chromatic-number upper bound, not optimality or a general greedy guarantee. Give (conflicting_edges,count,proper?), with edges in normalized sorted order.

View answer and explanation([[0, 1]], 1, No)

Only same-colored endpoints joined by an edge are conflicts. Reusing a color on nonadjacent vertices is allowed. The exact result, retaining the requested order, is ([[0, 1]], 1, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r05-graph-coloring-0040/)

Graph coloring · qf-mod31-coloring-conflict · candidate





## 41. List Every Monochromatic Edge, Not Every Repeated Color

Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5]]. Normalize each edge increasingly, then sort edges lexicographically; reversed duplicates and loops are forbidden. Keep isolated vertices in the carrier. No directed-graph conclusion is inferred. Total vertex colors=[15, 15, 16, 15, 16, 15] are labels, not cardinalities. Only adjacent endpoints must differ; nonadjacent vertices may share a color. A proper k-coloring uses at most k distinct labels, even if their numerical names exceed k−1. One such coloring gives only a chromatic-number upper bound, not optimality or a general greedy guarantee. Give (conflicting_edges,count,proper?), with edges in normalized sorted order.

View answer and explanation([[0, 1]], 1, No)

Only same-colored endpoints joined by an edge are conflicts. Reusing a color on nonadjacent vertices is allowed. The exact result, retaining the requested order, is ([[0, 1]], 1, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r05-graph-coloring-0041/)

Graph coloring · qf-mod31-coloring-conflict · candidate





## 42. List Every Monochromatic Edge, Not Every Repeated Color

Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5]]. Normalize each edge increasingly, then sort edges lexicographically; reversed duplicates and loops are forbidden. Keep isolated vertices in the carrier. No directed-graph conclusion is inferred. Total vertex colors=[16, 16, 17, 16, 17, 16] are labels, not cardinalities. Only adjacent endpoints must differ; nonadjacent vertices may share a color. A proper k-coloring uses at most k distinct labels, even if their numerical names exceed k−1. One such coloring gives only a chromatic-number upper bound, not optimality or a general greedy guarantee. Give (conflicting_edges,count,proper?), with edges in normalized sorted order.

View answer and explanation([[0, 1]], 1, No)

Only same-colored endpoints joined by an edge are conflicts. Reusing a color on nonadjacent vertices is allowed. The exact result, retaining the requested order, is ([[0, 1]], 1, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r05-graph-coloring-0042/)

Graph coloring · qf-mod31-coloring-conflict · candidate





## 43. List Every Monochromatic Edge, Not Every Repeated Color

Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5]]. Normalize each edge increasingly, then sort edges lexicographically; reversed duplicates and loops are forbidden. Keep isolated vertices in the carrier. No directed-graph conclusion is inferred. Total vertex colors=[17, 17, 18, 17, 18, 17] are labels, not cardinalities. Only adjacent endpoints must differ; nonadjacent vertices may share a color. A proper k-coloring uses at most k distinct labels, even if their numerical names exceed k−1. One such coloring gives only a chromatic-number upper bound, not optimality or a general greedy guarantee. Give (conflicting_edges,count,proper?), with edges in normalized sorted order.

View answer and explanation([[0, 1]], 1, No)

Only same-colored endpoints joined by an edge are conflicts. Reusing a color on nonadjacent vertices is allowed. The exact result, retaining the requested order, is ([[0, 1]], 1, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r05-graph-coloring-0043/)

Graph coloring · qf-mod31-coloring-conflict · candidate





## 44. List Every Monochromatic Edge, Not Every Repeated Color

Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5]]. Normalize each edge increasingly, then sort edges lexicographically; reversed duplicates and loops are forbidden. Keep isolated vertices in the carrier. No directed-graph conclusion is inferred. Total vertex colors=[18, 18, 19, 18, 19, 18] are labels, not cardinalities. Only adjacent endpoints must differ; nonadjacent vertices may share a color. A proper k-coloring uses at most k distinct labels, even if their numerical names exceed k−1. One such coloring gives only a chromatic-number upper bound, not optimality or a general greedy guarantee. Give (conflicting_edges,count,proper?), with edges in normalized sorted order.

View answer and explanation([[0, 1]], 1, No)

Only same-colored endpoints joined by an edge are conflicts. Reusing a color on nonadjacent vertices is allowed. The exact result, retaining the requested order, is ([[0, 1]], 1, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r05-graph-coloring-0044/)

Graph coloring · qf-mod31-coloring-conflict · candidate





## 45. List Every Monochromatic Edge, Not Every Repeated Color

Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5]]. Normalize each edge increasingly, then sort edges lexicographically; reversed duplicates and loops are forbidden. Keep isolated vertices in the carrier. No directed-graph conclusion is inferred. Total vertex colors=[19, 19, 20, 19, 20, 19] are labels, not cardinalities. Only adjacent endpoints must differ; nonadjacent vertices may share a color. A proper k-coloring uses at most k distinct labels, even if their numerical names exceed k−1. One such coloring gives only a chromatic-number upper bound, not optimality or a general greedy guarantee. Give (conflicting_edges,count,proper?), with edges in normalized sorted order.

View answer and explanation([[0, 1]], 1, No)

Only same-colored endpoints joined by an edge are conflicts. Reusing a color on nonadjacent vertices is allowed. The exact result, retaining the requested order, is ([[0, 1]], 1, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r05-graph-coloring-0045/)

Graph coloring · qf-mod31-coloring-conflict · candidate





## 46. List Every Monochromatic Edge, Not Every Repeated Color

Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5]]. Normalize each edge increasingly, then sort edges lexicographically; reversed duplicates and loops are forbidden. Keep isolated vertices in the carrier. No directed-graph conclusion is inferred. Total vertex colors=[20, 20, 21, 20, 21, 20] are labels, not cardinalities. Only adjacent endpoints must differ; nonadjacent vertices may share a color. A proper k-coloring uses at most k distinct labels, even if their numerical names exceed k−1. One such coloring gives only a chromatic-number upper bound, not optimality or a general greedy guarantee. Give (conflicting_edges,count,proper?), with edges in normalized sorted order.

View answer and explanation([[0, 1]], 1, No)

Only same-colored endpoints joined by an edge are conflicts. Reusing a color on nonadjacent vertices is allowed. The exact result, retaining the requested order, is ([[0, 1]], 1, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r05-graph-coloring-0046/)

Graph coloring · qf-mod31-coloring-conflict · candidate





## 47. List Every Monochromatic Edge, Not Every Repeated Color

Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5]]. Normalize each edge increasingly, then sort edges lexicographically; reversed duplicates and loops are forbidden. Keep isolated vertices in the carrier. No directed-graph conclusion is inferred. Total vertex colors=[21, 21, 22, 21, 22, 21] are labels, not cardinalities. Only adjacent endpoints must differ; nonadjacent vertices may share a color. A proper k-coloring uses at most k distinct labels, even if their numerical names exceed k−1. One such coloring gives only a chromatic-number upper bound, not optimality or a general greedy guarantee. Give (conflicting_edges,count,proper?), with edges in normalized sorted order.

View answer and explanation([[0, 1]], 1, No)

Only same-colored endpoints joined by an edge are conflicts. Reusing a color on nonadjacent vertices is allowed. The exact result, retaining the requested order, is ([[0, 1]], 1, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r05-graph-coloring-0047/)

Graph coloring · qf-mod31-coloring-conflict · candidate





## 48. List Every Monochromatic Edge, Not Every Repeated Color

Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5]]. Normalize each edge increasingly, then sort edges lexicographically; reversed duplicates and loops are forbidden. Keep isolated vertices in the carrier. No directed-graph conclusion is inferred. Total vertex colors=[22, 22, 23, 22, 23, 22] are labels, not cardinalities. Only adjacent endpoints must differ; nonadjacent vertices may share a color. A proper k-coloring uses at most k distinct labels, even if their numerical names exceed k−1. One such coloring gives only a chromatic-number upper bound, not optimality or a general greedy guarantee. Give (conflicting_edges,count,proper?), with edges in normalized sorted order.

View answer and explanation([[0, 1]], 1, No)

Only same-colored endpoints joined by an edge are conflicts. Reusing a color on nonadjacent vertices is allowed. The exact result, retaining the requested order, is ([[0, 1]], 1, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r05-graph-coloring-0048/)

Graph coloring · qf-mod31-coloring-conflict · candidate





## 49. List Every Monochromatic Edge, Not Every Repeated Color

Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5]]. Normalize each edge increasingly, then sort edges lexicographically; reversed duplicates and loops are forbidden. Keep isolated vertices in the carrier. No directed-graph conclusion is inferred. Total vertex colors=[23, 23, 24, 23, 24, 23] are labels, not cardinalities. Only adjacent endpoints must differ; nonadjacent vertices may share a color. A proper k-coloring uses at most k distinct labels, even if their numerical names exceed k−1. One such coloring gives only a chromatic-number upper bound, not optimality or a general greedy guarantee. Give (conflicting_edges,count,proper?), with edges in normalized sorted order.

View answer and explanation([[0, 1]], 1, No)

Only same-colored endpoints joined by an edge are conflicts. Reusing a color on nonadjacent vertices is allowed. The exact result, retaining the requested order, is ([[0, 1]], 1, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r05-graph-coloring-0049/)

Graph coloring · qf-mod31-coloring-conflict · candidate





## 50. List Every Monochromatic Edge, Not Every Repeated Color

Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5]]. Normalize each edge increasingly, then sort edges lexicographically; reversed duplicates and loops are forbidden. Keep isolated vertices in the carrier. No directed-graph conclusion is inferred. Total vertex colors=[24, 24, 25, 24, 25, 24] are labels, not cardinalities. Only adjacent endpoints must differ; nonadjacent vertices may share a color. A proper k-coloring uses at most k distinct labels, even if their numerical names exceed k−1. One such coloring gives only a chromatic-number upper bound, not optimality or a general greedy guarantee. Give (conflicting_edges,count,proper?), with edges in normalized sorted order.

View answer and explanation([[0, 1]], 1, No)

Only same-colored endpoints joined by an edge are conflicts. Reusing a color on nonadjacent vertices is allowed. The exact result, retaining the requested order, is ([[0, 1]], 1, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r05-graph-coloring-0050/)

Graph coloring · qf-mod31-coloring-conflict · candidate





## 51. A Nonoptimal Valid Coloring Still Supplies an Upper Bound

Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5]]. Normalize each edge increasingly, then sort edges lexicographically; reversed duplicates and loops are forbidden. Keep isolated vertices in the carrier. No directed-graph conclusion is inferred. Total vertex colors=[0, 1, 0, 1, 0, 1] are labels, not cardinalities. Only adjacent endpoints must differ; nonadjacent vertices may share a color. A proper k-coloring uses at most k distinct labels, even if their numerical names exceed k−1. One such coloring gives only a chromatic-number upper bound, not optimality or a general greedy guarantee. Proposed k=2. Give (used_count,k,proper?,certificate?,conclusion,upper_bound_not_exact_optimum). A valid certificate yields chromatic_number_at_most_k with the supplied numeric k; otherwise no_upper_bound_from_this_coloring.

View answer and explanation(2, 2, Yes, Yes, chromatic_number_at_most_2, upper_bound_not_exact_optimum)

Conjoin all-edge properness with used_count≤k. A loose k may be sufficient without being optimal, and failure of this coloring gives no general lower-bound certificate. The exact result, retaining the requested order, is (2, 2, Yes, Yes, chromatic_number_at_most_2, upper_bound_not_exact_optimum).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r05-graph-coloring-0051/)

Graph coloring · qf-mod31-coloring-upper-bound · candidate





## 52. A Nonoptimal Valid Coloring Still Supplies an Upper Bound

Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5]]. Normalize each edge increasingly, then sort edges lexicographically; reversed duplicates and loops are forbidden. Keep isolated vertices in the carrier. No directed-graph conclusion is inferred. Total vertex colors=[1, 2, 1, 2, 1, 2] are labels, not cardinalities. Only adjacent endpoints must differ; nonadjacent vertices may share a color. A proper k-coloring uses at most k distinct labels, even if their numerical names exceed k−1. One such coloring gives only a chromatic-number upper bound, not optimality or a general greedy guarantee. Proposed k=3. Give (used_count,k,proper?,certificate?,conclusion,upper_bound_not_exact_optimum). A valid certificate yields chromatic_number_at_most_k with the supplied numeric k; otherwise no_upper_bound_from_this_coloring.

View answer and explanation(2, 3, Yes, Yes, chromatic_number_at_most_3, upper_bound_not_exact_optimum)

Conjoin all-edge properness with used_count≤k. A loose k may be sufficient without being optimal, and failure of this coloring gives no general lower-bound certificate. The exact result, retaining the requested order, is (2, 3, Yes, Yes, chromatic_number_at_most_3, upper_bound_not_exact_optimum).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r05-graph-coloring-0052/)

Graph coloring · qf-mod31-coloring-upper-bound · candidate





## 53. A Nonoptimal Valid Coloring Still Supplies an Upper Bound

Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5]]. Normalize each edge increasingly, then sort edges lexicographically; reversed duplicates and loops are forbidden. Keep isolated vertices in the carrier. No directed-graph conclusion is inferred. Total vertex colors=[2, 3, 2, 3, 2, 3] are labels, not cardinalities. Only adjacent endpoints must differ; nonadjacent vertices may share a color. A proper k-coloring uses at most k distinct labels, even if their numerical names exceed k−1. One such coloring gives only a chromatic-number upper bound, not optimality or a general greedy guarantee. Proposed k=4. Give (used_count,k,proper?,certificate?,conclusion,upper_bound_not_exact_optimum). A valid certificate yields chromatic_number_at_most_k with the supplied numeric k; otherwise no_upper_bound_from_this_coloring.

View answer and explanation(2, 4, Yes, Yes, chromatic_number_at_most_4, upper_bound_not_exact_optimum)

Conjoin all-edge properness with used_count≤k. A loose k may be sufficient without being optimal, and failure of this coloring gives no general lower-bound certificate. The exact result, retaining the requested order, is (2, 4, Yes, Yes, chromatic_number_at_most_4, upper_bound_not_exact_optimum).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r05-graph-coloring-0053/)

Graph coloring · qf-mod31-coloring-upper-bound · candidate





## 54. A Nonoptimal Valid Coloring Still Supplies an Upper Bound

Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5]]. Normalize each edge increasingly, then sort edges lexicographically; reversed duplicates and loops are forbidden. Keep isolated vertices in the carrier. No directed-graph conclusion is inferred. Total vertex colors=[3, 4, 3, 4, 3, 4] are labels, not cardinalities. Only adjacent endpoints must differ; nonadjacent vertices may share a color. A proper k-coloring uses at most k distinct labels, even if their numerical names exceed k−1. One such coloring gives only a chromatic-number upper bound, not optimality or a general greedy guarantee. Proposed k=2. Give (used_count,k,proper?,certificate?,conclusion,upper_bound_not_exact_optimum). A valid certificate yields chromatic_number_at_most_k with the supplied numeric k; otherwise no_upper_bound_from_this_coloring.

View answer and explanation(2, 2, Yes, Yes, chromatic_number_at_most_2, upper_bound_not_exact_optimum)

Conjoin all-edge properness with used_count≤k. A loose k may be sufficient without being optimal, and failure of this coloring gives no general lower-bound certificate. The exact result, retaining the requested order, is (2, 2, Yes, Yes, chromatic_number_at_most_2, upper_bound_not_exact_optimum).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r05-graph-coloring-0054/)

Graph coloring · qf-mod31-coloring-upper-bound · candidate





## 55. A Nonoptimal Valid Coloring Still Supplies an Upper Bound

Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5]]. Normalize each edge increasingly, then sort edges lexicographically; reversed duplicates and loops are forbidden. Keep isolated vertices in the carrier. No directed-graph conclusion is inferred. Total vertex colors=[4, 5, 4, 5, 4, 5] are labels, not cardinalities. Only adjacent endpoints must differ; nonadjacent vertices may share a color. A proper k-coloring uses at most k distinct labels, even if their numerical names exceed k−1. One such coloring gives only a chromatic-number upper bound, not optimality or a general greedy guarantee. Proposed k=3. Give (used_count,k,proper?,certificate?,conclusion,upper_bound_not_exact_optimum). A valid certificate yields chromatic_number_at_most_k with the supplied numeric k; otherwise no_upper_bound_from_this_coloring.

View answer and explanation(2, 3, Yes, Yes, chromatic_number_at_most_3, upper_bound_not_exact_optimum)

Conjoin all-edge properness with used_count≤k. A loose k may be sufficient without being optimal, and failure of this coloring gives no general lower-bound certificate. The exact result, retaining the requested order, is (2, 3, Yes, Yes, chromatic_number_at_most_3, upper_bound_not_exact_optimum).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r05-graph-coloring-0055/)

Graph coloring · qf-mod31-coloring-upper-bound · candidate





## 56. A Nonoptimal Valid Coloring Still Supplies an Upper Bound

Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5]]. Normalize each edge increasingly, then sort edges lexicographically; reversed duplicates and loops are forbidden. Keep isolated vertices in the carrier. No directed-graph conclusion is inferred. Total vertex colors=[5, 6, 5, 6, 5, 6] are labels, not cardinalities. Only adjacent endpoints must differ; nonadjacent vertices may share a color. A proper k-coloring uses at most k distinct labels, even if their numerical names exceed k−1. One such coloring gives only a chromatic-number upper bound, not optimality or a general greedy guarantee. Proposed k=4. Give (used_count,k,proper?,certificate?,conclusion,upper_bound_not_exact_optimum). A valid certificate yields chromatic_number_at_most_k with the supplied numeric k; otherwise no_upper_bound_from_this_coloring.

View answer and explanation(2, 4, Yes, Yes, chromatic_number_at_most_4, upper_bound_not_exact_optimum)

Conjoin all-edge properness with used_count≤k. A loose k may be sufficient without being optimal, and failure of this coloring gives no general lower-bound certificate. The exact result, retaining the requested order, is (2, 4, Yes, Yes, chromatic_number_at_most_4, upper_bound_not_exact_optimum).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r05-graph-coloring-0056/)

Graph coloring · qf-mod31-coloring-upper-bound · candidate





## 57. A Nonoptimal Valid Coloring Still Supplies an Upper Bound

Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5]]. Normalize each edge increasingly, then sort edges lexicographically; reversed duplicates and loops are forbidden. Keep isolated vertices in the carrier. No directed-graph conclusion is inferred. Total vertex colors=[6, 7, 6, 7, 6, 7] are labels, not cardinalities. Only adjacent endpoints must differ; nonadjacent vertices may share a color. A proper k-coloring uses at most k distinct labels, even if their numerical names exceed k−1. One such coloring gives only a chromatic-number upper bound, not optimality or a general greedy guarantee. Proposed k=2. Give (used_count,k,proper?,certificate?,conclusion,upper_bound_not_exact_optimum). A valid certificate yields chromatic_number_at_most_k with the supplied numeric k; otherwise no_upper_bound_from_this_coloring.

View answer and explanation(2, 2, Yes, Yes, chromatic_number_at_most_2, upper_bound_not_exact_optimum)

Conjoin all-edge properness with used_count≤k. A loose k may be sufficient without being optimal, and failure of this coloring gives no general lower-bound certificate. The exact result, retaining the requested order, is (2, 2, Yes, Yes, chromatic_number_at_most_2, upper_bound_not_exact_optimum).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r05-graph-coloring-0057/)

Graph coloring · qf-mod31-coloring-upper-bound · candidate





## 58. A Nonoptimal Valid Coloring Still Supplies an Upper Bound

Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5]]. Normalize each edge increasingly, then sort edges lexicographically; reversed duplicates and loops are forbidden. Keep isolated vertices in the carrier. No directed-graph conclusion is inferred. Total vertex colors=[7, 8, 7, 8, 7, 8] are labels, not cardinalities. Only adjacent endpoints must differ; nonadjacent vertices may share a color. A proper k-coloring uses at most k distinct labels, even if their numerical names exceed k−1. One such coloring gives only a chromatic-number upper bound, not optimality or a general greedy guarantee. Proposed k=3. Give (used_count,k,proper?,certificate?,conclusion,upper_bound_not_exact_optimum). A valid certificate yields chromatic_number_at_most_k with the supplied numeric k; otherwise no_upper_bound_from_this_coloring.

View answer and explanation(2, 3, Yes, Yes, chromatic_number_at_most_3, upper_bound_not_exact_optimum)

Conjoin all-edge properness with used_count≤k. A loose k may be sufficient without being optimal, and failure of this coloring gives no general lower-bound certificate. The exact result, retaining the requested order, is (2, 3, Yes, Yes, chromatic_number_at_most_3, upper_bound_not_exact_optimum).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r05-graph-coloring-0058/)

Graph coloring · qf-mod31-coloring-upper-bound · candidate





## 59. A Nonoptimal Valid Coloring Still Supplies an Upper Bound

Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5]]. Normalize each edge increasingly, then sort edges lexicographically; reversed duplicates and loops are forbidden. Keep isolated vertices in the carrier. No directed-graph conclusion is inferred. Total vertex colors=[8, 9, 8, 9, 8, 9] are labels, not cardinalities. Only adjacent endpoints must differ; nonadjacent vertices may share a color. A proper k-coloring uses at most k distinct labels, even if their numerical names exceed k−1. One such coloring gives only a chromatic-number upper bound, not optimality or a general greedy guarantee. Proposed k=4. Give (used_count,k,proper?,certificate?,conclusion,upper_bound_not_exact_optimum). A valid certificate yields chromatic_number_at_most_k with the supplied numeric k; otherwise no_upper_bound_from_this_coloring.

View answer and explanation(2, 4, Yes, Yes, chromatic_number_at_most_4, upper_bound_not_exact_optimum)

Conjoin all-edge properness with used_count≤k. A loose k may be sufficient without being optimal, and failure of this coloring gives no general lower-bound certificate. The exact result, retaining the requested order, is (2, 4, Yes, Yes, chromatic_number_at_most_4, upper_bound_not_exact_optimum).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r05-graph-coloring-0059/)

Graph coloring · qf-mod31-coloring-upper-bound · candidate





## 60. A Nonoptimal Valid Coloring Still Supplies an Upper Bound

Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5]]. Normalize each edge increasingly, then sort edges lexicographically; reversed duplicates and loops are forbidden. Keep isolated vertices in the carrier. No directed-graph conclusion is inferred. Total vertex colors=[9, 10, 9, 10, 9, 10] are labels, not cardinalities. Only adjacent endpoints must differ; nonadjacent vertices may share a color. A proper k-coloring uses at most k distinct labels, even if their numerical names exceed k−1. One such coloring gives only a chromatic-number upper bound, not optimality or a general greedy guarantee. Proposed k=2. Give (used_count,k,proper?,certificate?,conclusion,upper_bound_not_exact_optimum). A valid certificate yields chromatic_number_at_most_k with the supplied numeric k; otherwise no_upper_bound_from_this_coloring.

View answer and explanation(2, 2, Yes, Yes, chromatic_number_at_most_2, upper_bound_not_exact_optimum)

Conjoin all-edge properness with used_count≤k. A loose k may be sufficient without being optimal, and failure of this coloring gives no general lower-bound certificate. The exact result, retaining the requested order, is (2, 2, Yes, Yes, chromatic_number_at_most_2, upper_bound_not_exact_optimum).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r05-graph-coloring-0060/)

Graph coloring · qf-mod31-coloring-upper-bound · candidate





## 61. A Nonoptimal Valid Coloring Still Supplies an Upper Bound

Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5]]. Normalize each edge increasingly, then sort edges lexicographically; reversed duplicates and loops are forbidden. Keep isolated vertices in the carrier. No directed-graph conclusion is inferred. Total vertex colors=[10, 11, 10, 11, 10, 11] are labels, not cardinalities. Only adjacent endpoints must differ; nonadjacent vertices may share a color. A proper k-coloring uses at most k distinct labels, even if their numerical names exceed k−1. One such coloring gives only a chromatic-number upper bound, not optimality or a general greedy guarantee. Proposed k=3. Give (used_count,k,proper?,certificate?,conclusion,upper_bound_not_exact_optimum). A valid certificate yields chromatic_number_at_most_k with the supplied numeric k; otherwise no_upper_bound_from_this_coloring.

View answer and explanation(2, 3, Yes, Yes, chromatic_number_at_most_3, upper_bound_not_exact_optimum)

Conjoin all-edge properness with used_count≤k. A loose k may be sufficient without being optimal, and failure of this coloring gives no general lower-bound certificate. The exact result, retaining the requested order, is (2, 3, Yes, Yes, chromatic_number_at_most_3, upper_bound_not_exact_optimum).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r05-graph-coloring-0061/)

Graph coloring · qf-mod31-coloring-upper-bound · candidate





## 62. A Nonoptimal Valid Coloring Still Supplies an Upper Bound

Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5]]. Normalize each edge increasingly, then sort edges lexicographically; reversed duplicates and loops are forbidden. Keep isolated vertices in the carrier. No directed-graph conclusion is inferred. Total vertex colors=[11, 12, 11, 12, 11, 12] are labels, not cardinalities. Only adjacent endpoints must differ; nonadjacent vertices may share a color. A proper k-coloring uses at most k distinct labels, even if their numerical names exceed k−1. One such coloring gives only a chromatic-number upper bound, not optimality or a general greedy guarantee. Proposed k=4. Give (used_count,k,proper?,certificate?,conclusion,upper_bound_not_exact_optimum). A valid certificate yields chromatic_number_at_most_k with the supplied numeric k; otherwise no_upper_bound_from_this_coloring.

View answer and explanation(2, 4, Yes, Yes, chromatic_number_at_most_4, upper_bound_not_exact_optimum)

Conjoin all-edge properness with used_count≤k. A loose k may be sufficient without being optimal, and failure of this coloring gives no general lower-bound certificate. The exact result, retaining the requested order, is (2, 4, Yes, Yes, chromatic_number_at_most_4, upper_bound_not_exact_optimum).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r05-graph-coloring-0062/)

Graph coloring · qf-mod31-coloring-upper-bound · candidate





## 63. A Nonoptimal Valid Coloring Still Supplies an Upper Bound

Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5]]. Normalize each edge increasingly, then sort edges lexicographically; reversed duplicates and loops are forbidden. Keep isolated vertices in the carrier. No directed-graph conclusion is inferred. Total vertex colors=[12, 13, 12, 13, 12, 13] are labels, not cardinalities. Only adjacent endpoints must differ; nonadjacent vertices may share a color. A proper k-coloring uses at most k distinct labels, even if their numerical names exceed k−1. One such coloring gives only a chromatic-number upper bound, not optimality or a general greedy guarantee. Proposed k=2. Give (used_count,k,proper?,certificate?,conclusion,upper_bound_not_exact_optimum). A valid certificate yields chromatic_number_at_most_k with the supplied numeric k; otherwise no_upper_bound_from_this_coloring.

View answer and explanation(2, 2, Yes, Yes, chromatic_number_at_most_2, upper_bound_not_exact_optimum)

Conjoin all-edge properness with used_count≤k. A loose k may be sufficient without being optimal, and failure of this coloring gives no general lower-bound certificate. The exact result, retaining the requested order, is (2, 2, Yes, Yes, chromatic_number_at_most_2, upper_bound_not_exact_optimum).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r05-graph-coloring-0063/)

Graph coloring · qf-mod31-coloring-upper-bound · candidate





## 64. A Nonoptimal Valid Coloring Still Supplies an Upper Bound

Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5]]. Normalize each edge increasingly, then sort edges lexicographically; reversed duplicates and loops are forbidden. Keep isolated vertices in the carrier. No directed-graph conclusion is inferred. Total vertex colors=[13, 14, 13, 14, 13, 14] are labels, not cardinalities. Only adjacent endpoints must differ; nonadjacent vertices may share a color. A proper k-coloring uses at most k distinct labels, even if their numerical names exceed k−1. One such coloring gives only a chromatic-number upper bound, not optimality or a general greedy guarantee. Proposed k=3. Give (used_count,k,proper?,certificate?,conclusion,upper_bound_not_exact_optimum). A valid certificate yields chromatic_number_at_most_k with the supplied numeric k; otherwise no_upper_bound_from_this_coloring.

View answer and explanation(2, 3, Yes, Yes, chromatic_number_at_most_3, upper_bound_not_exact_optimum)

Conjoin all-edge properness with used_count≤k. A loose k may be sufficient without being optimal, and failure of this coloring gives no general lower-bound certificate. The exact result, retaining the requested order, is (2, 3, Yes, Yes, chromatic_number_at_most_3, upper_bound_not_exact_optimum).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r05-graph-coloring-0064/)

Graph coloring · qf-mod31-coloring-upper-bound · candidate





## 65. A Nonoptimal Valid Coloring Still Supplies an Upper Bound

Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5]]. Normalize each edge increasingly, then sort edges lexicographically; reversed duplicates and loops are forbidden. Keep isolated vertices in the carrier. No directed-graph conclusion is inferred. Total vertex colors=[14, 15, 14, 15, 14, 15] are labels, not cardinalities. Only adjacent endpoints must differ; nonadjacent vertices may share a color. A proper k-coloring uses at most k distinct labels, even if their numerical names exceed k−1. One such coloring gives only a chromatic-number upper bound, not optimality or a general greedy guarantee. Proposed k=4. Give (used_count,k,proper?,certificate?,conclusion,upper_bound_not_exact_optimum). A valid certificate yields chromatic_number_at_most_k with the supplied numeric k; otherwise no_upper_bound_from_this_coloring.

View answer and explanation(2, 4, Yes, Yes, chromatic_number_at_most_4, upper_bound_not_exact_optimum)

Conjoin all-edge properness with used_count≤k. A loose k may be sufficient without being optimal, and failure of this coloring gives no general lower-bound certificate. The exact result, retaining the requested order, is (2, 4, Yes, Yes, chromatic_number_at_most_4, upper_bound_not_exact_optimum).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r05-graph-coloring-0065/)

Graph coloring · qf-mod31-coloring-upper-bound · candidate





## 66. A Nonoptimal Valid Coloring Still Supplies an Upper Bound

Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5]]. Normalize each edge increasingly, then sort edges lexicographically; reversed duplicates and loops are forbidden. Keep isolated vertices in the carrier. No directed-graph conclusion is inferred. Total vertex colors=[15, 16, 15, 16, 15, 16] are labels, not cardinalities. Only adjacent endpoints must differ; nonadjacent vertices may share a color. A proper k-coloring uses at most k distinct labels, even if their numerical names exceed k−1. One such coloring gives only a chromatic-number upper bound, not optimality or a general greedy guarantee. Proposed k=2. Give (used_count,k,proper?,certificate?,conclusion,upper_bound_not_exact_optimum). A valid certificate yields chromatic_number_at_most_k with the supplied numeric k; otherwise no_upper_bound_from_this_coloring.

View answer and explanation(2, 2, Yes, Yes, chromatic_number_at_most_2, upper_bound_not_exact_optimum)

Conjoin all-edge properness with used_count≤k. A loose k may be sufficient without being optimal, and failure of this coloring gives no general lower-bound certificate. The exact result, retaining the requested order, is (2, 2, Yes, Yes, chromatic_number_at_most_2, upper_bound_not_exact_optimum).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r05-graph-coloring-0066/)

Graph coloring · qf-mod31-coloring-upper-bound · candidate





## 67. A Nonoptimal Valid Coloring Still Supplies an Upper Bound

Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5]]. Normalize each edge increasingly, then sort edges lexicographically; reversed duplicates and loops are forbidden. Keep isolated vertices in the carrier. No directed-graph conclusion is inferred. Total vertex colors=[16, 17, 16, 17, 16, 17] are labels, not cardinalities. Only adjacent endpoints must differ; nonadjacent vertices may share a color. A proper k-coloring uses at most k distinct labels, even if their numerical names exceed k−1. One such coloring gives only a chromatic-number upper bound, not optimality or a general greedy guarantee. Proposed k=3. Give (used_count,k,proper?,certificate?,conclusion,upper_bound_not_exact_optimum). A valid certificate yields chromatic_number_at_most_k with the supplied numeric k; otherwise no_upper_bound_from_this_coloring.

View answer and explanation(2, 3, Yes, Yes, chromatic_number_at_most_3, upper_bound_not_exact_optimum)

Conjoin all-edge properness with used_count≤k. A loose k may be sufficient without being optimal, and failure of this coloring gives no general lower-bound certificate. The exact result, retaining the requested order, is (2, 3, Yes, Yes, chromatic_number_at_most_3, upper_bound_not_exact_optimum).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r05-graph-coloring-0067/)

Graph coloring · qf-mod31-coloring-upper-bound · candidate





## 68. A Nonoptimal Valid Coloring Still Supplies an Upper Bound

Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5]]. Normalize each edge increasingly, then sort edges lexicographically; reversed duplicates and loops are forbidden. Keep isolated vertices in the carrier. No directed-graph conclusion is inferred. Total vertex colors=[17, 18, 17, 18, 17, 18] are labels, not cardinalities. Only adjacent endpoints must differ; nonadjacent vertices may share a color. A proper k-coloring uses at most k distinct labels, even if their numerical names exceed k−1. One such coloring gives only a chromatic-number upper bound, not optimality or a general greedy guarantee. Proposed k=4. Give (used_count,k,proper?,certificate?,conclusion,upper_bound_not_exact_optimum). A valid certificate yields chromatic_number_at_most_k with the supplied numeric k; otherwise no_upper_bound_from_this_coloring.

View answer and explanation(2, 4, Yes, Yes, chromatic_number_at_most_4, upper_bound_not_exact_optimum)

Conjoin all-edge properness with used_count≤k. A loose k may be sufficient without being optimal, and failure of this coloring gives no general lower-bound certificate. The exact result, retaining the requested order, is (2, 4, Yes, Yes, chromatic_number_at_most_4, upper_bound_not_exact_optimum).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r05-graph-coloring-0068/)

Graph coloring · qf-mod31-coloring-upper-bound · candidate





## 69. A Nonoptimal Valid Coloring Still Supplies an Upper Bound

Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5]]. Normalize each edge increasingly, then sort edges lexicographically; reversed duplicates and loops are forbidden. Keep isolated vertices in the carrier. No directed-graph conclusion is inferred. Total vertex colors=[18, 19, 18, 19, 18, 19] are labels, not cardinalities. Only adjacent endpoints must differ; nonadjacent vertices may share a color. A proper k-coloring uses at most k distinct labels, even if their numerical names exceed k−1. One such coloring gives only a chromatic-number upper bound, not optimality or a general greedy guarantee. Proposed k=2. Give (used_count,k,proper?,certificate?,conclusion,upper_bound_not_exact_optimum). A valid certificate yields chromatic_number_at_most_k with the supplied numeric k; otherwise no_upper_bound_from_this_coloring.

View answer and explanation(2, 2, Yes, Yes, chromatic_number_at_most_2, upper_bound_not_exact_optimum)

Conjoin all-edge properness with used_count≤k. A loose k may be sufficient without being optimal, and failure of this coloring gives no general lower-bound certificate. The exact result, retaining the requested order, is (2, 2, Yes, Yes, chromatic_number_at_most_2, upper_bound_not_exact_optimum).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r05-graph-coloring-0069/)

Graph coloring · qf-mod31-coloring-upper-bound · candidate





## 70. A Nonoptimal Valid Coloring Still Supplies an Upper Bound

Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5]]. Normalize each edge increasingly, then sort edges lexicographically; reversed duplicates and loops are forbidden. Keep isolated vertices in the carrier. No directed-graph conclusion is inferred. Total vertex colors=[19, 20, 19, 20, 19, 20] are labels, not cardinalities. Only adjacent endpoints must differ; nonadjacent vertices may share a color. A proper k-coloring uses at most k distinct labels, even if their numerical names exceed k−1. One such coloring gives only a chromatic-number upper bound, not optimality or a general greedy guarantee. Proposed k=3. Give (used_count,k,proper?,certificate?,conclusion,upper_bound_not_exact_optimum). A valid certificate yields chromatic_number_at_most_k with the supplied numeric k; otherwise no_upper_bound_from_this_coloring.

View answer and explanation(2, 3, Yes, Yes, chromatic_number_at_most_3, upper_bound_not_exact_optimum)

Conjoin all-edge properness with used_count≤k. A loose k may be sufficient without being optimal, and failure of this coloring gives no general lower-bound certificate. The exact result, retaining the requested order, is (2, 3, Yes, Yes, chromatic_number_at_most_3, upper_bound_not_exact_optimum).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r05-graph-coloring-0070/)

Graph coloring · qf-mod31-coloring-upper-bound · candidate





## 71. A Nonoptimal Valid Coloring Still Supplies an Upper Bound

Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5]]. Normalize each edge increasingly, then sort edges lexicographically; reversed duplicates and loops are forbidden. Keep isolated vertices in the carrier. No directed-graph conclusion is inferred. Total vertex colors=[20, 21, 20, 21, 20, 21] are labels, not cardinalities. Only adjacent endpoints must differ; nonadjacent vertices may share a color. A proper k-coloring uses at most k distinct labels, even if their numerical names exceed k−1. One such coloring gives only a chromatic-number upper bound, not optimality or a general greedy guarantee. Proposed k=4. Give (used_count,k,proper?,certificate?,conclusion,upper_bound_not_exact_optimum). A valid certificate yields chromatic_number_at_most_k with the supplied numeric k; otherwise no_upper_bound_from_this_coloring.

View answer and explanation(2, 4, Yes, Yes, chromatic_number_at_most_4, upper_bound_not_exact_optimum)

Conjoin all-edge properness with used_count≤k. A loose k may be sufficient without being optimal, and failure of this coloring gives no general lower-bound certificate. The exact result, retaining the requested order, is (2, 4, Yes, Yes, chromatic_number_at_most_4, upper_bound_not_exact_optimum).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r05-graph-coloring-0071/)

Graph coloring · qf-mod31-coloring-upper-bound · candidate





## 72. A Nonoptimal Valid Coloring Still Supplies an Upper Bound

Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5]]. Normalize each edge increasingly, then sort edges lexicographically; reversed duplicates and loops are forbidden. Keep isolated vertices in the carrier. No directed-graph conclusion is inferred. Total vertex colors=[21, 22, 21, 22, 21, 22] are labels, not cardinalities. Only adjacent endpoints must differ; nonadjacent vertices may share a color. A proper k-coloring uses at most k distinct labels, even if their numerical names exceed k−1. One such coloring gives only a chromatic-number upper bound, not optimality or a general greedy guarantee. Proposed k=2. Give (used_count,k,proper?,certificate?,conclusion,upper_bound_not_exact_optimum). A valid certificate yields chromatic_number_at_most_k with the supplied numeric k; otherwise no_upper_bound_from_this_coloring.

View answer and explanation(2, 2, Yes, Yes, chromatic_number_at_most_2, upper_bound_not_exact_optimum)

Conjoin all-edge properness with used_count≤k. A loose k may be sufficient without being optimal, and failure of this coloring gives no general lower-bound certificate. The exact result, retaining the requested order, is (2, 2, Yes, Yes, chromatic_number_at_most_2, upper_bound_not_exact_optimum).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r05-graph-coloring-0072/)

Graph coloring · qf-mod31-coloring-upper-bound · candidate





## 73. A Nonoptimal Valid Coloring Still Supplies an Upper Bound

Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5]]. Normalize each edge increasingly, then sort edges lexicographically; reversed duplicates and loops are forbidden. Keep isolated vertices in the carrier. No directed-graph conclusion is inferred. Total vertex colors=[22, 23, 22, 23, 22, 23] are labels, not cardinalities. Only adjacent endpoints must differ; nonadjacent vertices may share a color. A proper k-coloring uses at most k distinct labels, even if their numerical names exceed k−1. One such coloring gives only a chromatic-number upper bound, not optimality or a general greedy guarantee. Proposed k=3. Give (used_count,k,proper?,certificate?,conclusion,upper_bound_not_exact_optimum). A valid certificate yields chromatic_number_at_most_k with the supplied numeric k; otherwise no_upper_bound_from_this_coloring.

View answer and explanation(2, 3, Yes, Yes, chromatic_number_at_most_3, upper_bound_not_exact_optimum)

Conjoin all-edge properness with used_count≤k. A loose k may be sufficient without being optimal, and failure of this coloring gives no general lower-bound certificate. The exact result, retaining the requested order, is (2, 3, Yes, Yes, chromatic_number_at_most_3, upper_bound_not_exact_optimum).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r05-graph-coloring-0073/)

Graph coloring · qf-mod31-coloring-upper-bound · candidate





## 74. A Nonoptimal Valid Coloring Still Supplies an Upper Bound

Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5]]. Normalize each edge increasingly, then sort edges lexicographically; reversed duplicates and loops are forbidden. Keep isolated vertices in the carrier. No directed-graph conclusion is inferred. Total vertex colors=[23, 24, 23, 24, 23, 24] are labels, not cardinalities. Only adjacent endpoints must differ; nonadjacent vertices may share a color. A proper k-coloring uses at most k distinct labels, even if their numerical names exceed k−1. One such coloring gives only a chromatic-number upper bound, not optimality or a general greedy guarantee. Proposed k=4. Give (used_count,k,proper?,certificate?,conclusion,upper_bound_not_exact_optimum). A valid certificate yields chromatic_number_at_most_k with the supplied numeric k; otherwise no_upper_bound_from_this_coloring.

View answer and explanation(2, 4, Yes, Yes, chromatic_number_at_most_4, upper_bound_not_exact_optimum)

Conjoin all-edge properness with used_count≤k. A loose k may be sufficient without being optimal, and failure of this coloring gives no general lower-bound certificate. The exact result, retaining the requested order, is (2, 4, Yes, Yes, chromatic_number_at_most_4, upper_bound_not_exact_optimum).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r05-graph-coloring-0074/)

Graph coloring · qf-mod31-coloring-upper-bound · candidate





## 75. A Nonoptimal Valid Coloring Still Supplies an Upper Bound

Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5]]. Normalize each edge increasingly, then sort edges lexicographically; reversed duplicates and loops are forbidden. Keep isolated vertices in the carrier. No directed-graph conclusion is inferred. Total vertex colors=[24, 25, 24, 25, 24, 25] are labels, not cardinalities. Only adjacent endpoints must differ; nonadjacent vertices may share a color. A proper k-coloring uses at most k distinct labels, even if their numerical names exceed k−1. One such coloring gives only a chromatic-number upper bound, not optimality or a general greedy guarantee. Proposed k=2. Give (used_count,k,proper?,certificate?,conclusion,upper_bound_not_exact_optimum). A valid certificate yields chromatic_number_at_most_k with the supplied numeric k; otherwise no_upper_bound_from_this_coloring.

View answer and explanation(2, 2, Yes, Yes, chromatic_number_at_most_2, upper_bound_not_exact_optimum)

Conjoin all-edge properness with used_count≤k. A loose k may be sufficient without being optimal, and failure of this coloring gives no general lower-bound certificate. The exact result, retaining the requested order, is (2, 2, Yes, Yes, chromatic_number_at_most_2, upper_bound_not_exact_optimum).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r05-graph-coloring-0075/)

Graph coloring · qf-mod31-coloring-upper-bound · candidate





## 76. Injective Renaming of Used Colors Preserves the Vertex Partition

Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5]]. Normalize each edge increasingly, then sort edges lexicographically; reversed duplicates and loops are forbidden. Keep isolated vertices in the carrier. No directed-graph conclusion is inferred. Total vertex colors=[0, 1, 0, 1, 0, 1] are labels, not cardinalities. Only adjacent endpoints must differ; nonadjacent vertices may share a color. A proper k-coloring uses at most k distinct labels, even if their numerical names exceed k−1. One such coloring gives only a chromatic-number upper bound, not optimality or a general greedy guarantee. Color relabel table=[10, 40] is indexed by the original color labels and must be injective on the labels actually used. Give (colors,recolored,proper_before?,proper_after?,same_conflicts?,color_names_do_not_change_vertex_partition).

View answer and explanation([0, 1, 0, 1, 0, 1], [10, 40, 10, 40, 10, 40], Yes, Yes, Yes, color_names_do_not_change_vertex_partition)

Apply the table to every vertex in original order. Distinct used colors must remain distinct, so exactly the same edges remain monochromatic. Even the conflicts of a failed coloring are preserved, not repaired. The exact result, retaining the requested order, is ([0, 1, 0, 1, 0, 1], [10, 40, 10, 40, 10, 40], Yes, Yes, Yes, color_names_do_not_change_vertex_partition).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r05-graph-coloring-0076/)

Graph coloring · qf-mod31-coloring-relabel-invariance · candidate





## 77. Injective Renaming of Used Colors Preserves the Vertex Partition

Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5]]. Normalize each edge increasingly, then sort edges lexicographically; reversed duplicates and loops are forbidden. Keep isolated vertices in the carrier. No directed-graph conclusion is inferred. Total vertex colors=[0, 1, 0, 1, 0, 1] are labels, not cardinalities. Only adjacent endpoints must differ; nonadjacent vertices may share a color. A proper k-coloring uses at most k distinct labels, even if their numerical names exceed k−1. One such coloring gives only a chromatic-number upper bound, not optimality or a general greedy guarantee. Color relabel table=[11, 41] is indexed by the original color labels and must be injective on the labels actually used. Give (colors,recolored,proper_before?,proper_after?,same_conflicts?,color_names_do_not_change_vertex_partition).

View answer and explanation([0, 1, 0, 1, 0, 1], [11, 41, 11, 41, 11, 41], Yes, Yes, Yes, color_names_do_not_change_vertex_partition)

Apply the table to every vertex in original order. Distinct used colors must remain distinct, so exactly the same edges remain monochromatic. Even the conflicts of a failed coloring are preserved, not repaired. The exact result, retaining the requested order, is ([0, 1, 0, 1, 0, 1], [11, 41, 11, 41, 11, 41], Yes, Yes, Yes, color_names_do_not_change_vertex_partition).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r05-graph-coloring-0077/)

Graph coloring · qf-mod31-coloring-relabel-invariance · candidate





## 78. Injective Renaming of Used Colors Preserves the Vertex Partition

Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5]]. Normalize each edge increasingly, then sort edges lexicographically; reversed duplicates and loops are forbidden. Keep isolated vertices in the carrier. No directed-graph conclusion is inferred. Total vertex colors=[0, 1, 0, 1, 0, 1] are labels, not cardinalities. Only adjacent endpoints must differ; nonadjacent vertices may share a color. A proper k-coloring uses at most k distinct labels, even if their numerical names exceed k−1. One such coloring gives only a chromatic-number upper bound, not optimality or a general greedy guarantee. Color relabel table=[12, 42] is indexed by the original color labels and must be injective on the labels actually used. Give (colors,recolored,proper_before?,proper_after?,same_conflicts?,color_names_do_not_change_vertex_partition).

View answer and explanation([0, 1, 0, 1, 0, 1], [12, 42, 12, 42, 12, 42], Yes, Yes, Yes, color_names_do_not_change_vertex_partition)

Apply the table to every vertex in original order. Distinct used colors must remain distinct, so exactly the same edges remain monochromatic. Even the conflicts of a failed coloring are preserved, not repaired. The exact result, retaining the requested order, is ([0, 1, 0, 1, 0, 1], [12, 42, 12, 42, 12, 42], Yes, Yes, Yes, color_names_do_not_change_vertex_partition).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r05-graph-coloring-0078/)

Graph coloring · qf-mod31-coloring-relabel-invariance · candidate





## 79. Injective Renaming of Used Colors Preserves the Vertex Partition

Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5]]. Normalize each edge increasingly, then sort edges lexicographically; reversed duplicates and loops are forbidden. Keep isolated vertices in the carrier. No directed-graph conclusion is inferred. Total vertex colors=[0, 1, 0, 1, 0, 1] are labels, not cardinalities. Only adjacent endpoints must differ; nonadjacent vertices may share a color. A proper k-coloring uses at most k distinct labels, even if their numerical names exceed k−1. One such coloring gives only a chromatic-number upper bound, not optimality or a general greedy guarantee. Color relabel table=[13, 43] is indexed by the original color labels and must be injective on the labels actually used. Give (colors,recolored,proper_before?,proper_after?,same_conflicts?,color_names_do_not_change_vertex_partition).

View answer and explanation([0, 1, 0, 1, 0, 1], [13, 43, 13, 43, 13, 43], Yes, Yes, Yes, color_names_do_not_change_vertex_partition)

Apply the table to every vertex in original order. Distinct used colors must remain distinct, so exactly the same edges remain monochromatic. Even the conflicts of a failed coloring are preserved, not repaired. The exact result, retaining the requested order, is ([0, 1, 0, 1, 0, 1], [13, 43, 13, 43, 13, 43], Yes, Yes, Yes, color_names_do_not_change_vertex_partition).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r05-graph-coloring-0079/)

Graph coloring · qf-mod31-coloring-relabel-invariance · candidate





## 80. Injective Renaming of Used Colors Preserves the Vertex Partition

Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5]]. Normalize each edge increasingly, then sort edges lexicographically; reversed duplicates and loops are forbidden. Keep isolated vertices in the carrier. No directed-graph conclusion is inferred. Total vertex colors=[0, 1, 0, 1, 0, 1] are labels, not cardinalities. Only adjacent endpoints must differ; nonadjacent vertices may share a color. A proper k-coloring uses at most k distinct labels, even if their numerical names exceed k−1. One such coloring gives only a chromatic-number upper bound, not optimality or a general greedy guarantee. Color relabel table=[14, 44] is indexed by the original color labels and must be injective on the labels actually used. Give (colors,recolored,proper_before?,proper_after?,same_conflicts?,color_names_do_not_change_vertex_partition).

View answer and explanation([0, 1, 0, 1, 0, 1], [14, 44, 14, 44, 14, 44], Yes, Yes, Yes, color_names_do_not_change_vertex_partition)

Apply the table to every vertex in original order. Distinct used colors must remain distinct, so exactly the same edges remain monochromatic. Even the conflicts of a failed coloring are preserved, not repaired. The exact result, retaining the requested order, is ([0, 1, 0, 1, 0, 1], [14, 44, 14, 44, 14, 44], Yes, Yes, Yes, color_names_do_not_change_vertex_partition).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r05-graph-coloring-0080/)

Graph coloring · qf-mod31-coloring-relabel-invariance · candidate





## 81. Injective Renaming of Used Colors Preserves the Vertex Partition

Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5]]. Normalize each edge increasingly, then sort edges lexicographically; reversed duplicates and loops are forbidden. Keep isolated vertices in the carrier. No directed-graph conclusion is inferred. Total vertex colors=[0, 1, 0, 1, 0, 1] are labels, not cardinalities. Only adjacent endpoints must differ; nonadjacent vertices may share a color. A proper k-coloring uses at most k distinct labels, even if their numerical names exceed k−1. One such coloring gives only a chromatic-number upper bound, not optimality or a general greedy guarantee. Color relabel table=[15, 45] is indexed by the original color labels and must be injective on the labels actually used. Give (colors,recolored,proper_before?,proper_after?,same_conflicts?,color_names_do_not_change_vertex_partition).

View answer and explanation([0, 1, 0, 1, 0, 1], [15, 45, 15, 45, 15, 45], Yes, Yes, Yes, color_names_do_not_change_vertex_partition)

Apply the table to every vertex in original order. Distinct used colors must remain distinct, so exactly the same edges remain monochromatic. Even the conflicts of a failed coloring are preserved, not repaired. The exact result, retaining the requested order, is ([0, 1, 0, 1, 0, 1], [15, 45, 15, 45, 15, 45], Yes, Yes, Yes, color_names_do_not_change_vertex_partition).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r05-graph-coloring-0081/)

Graph coloring · qf-mod31-coloring-relabel-invariance · candidate





## 82. Injective Renaming of Used Colors Preserves the Vertex Partition

Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5]]. Normalize each edge increasingly, then sort edges lexicographically; reversed duplicates and loops are forbidden. Keep isolated vertices in the carrier. No directed-graph conclusion is inferred. Total vertex colors=[0, 1, 0, 1, 0, 1] are labels, not cardinalities. Only adjacent endpoints must differ; nonadjacent vertices may share a color. A proper k-coloring uses at most k distinct labels, even if their numerical names exceed k−1. One such coloring gives only a chromatic-number upper bound, not optimality or a general greedy guarantee. Color relabel table=[16, 46] is indexed by the original color labels and must be injective on the labels actually used. Give (colors,recolored,proper_before?,proper_after?,same_conflicts?,color_names_do_not_change_vertex_partition).

View answer and explanation([0, 1, 0, 1, 0, 1], [16, 46, 16, 46, 16, 46], Yes, Yes, Yes, color_names_do_not_change_vertex_partition)

Apply the table to every vertex in original order. Distinct used colors must remain distinct, so exactly the same edges remain monochromatic. Even the conflicts of a failed coloring are preserved, not repaired. The exact result, retaining the requested order, is ([0, 1, 0, 1, 0, 1], [16, 46, 16, 46, 16, 46], Yes, Yes, Yes, color_names_do_not_change_vertex_partition).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r05-graph-coloring-0082/)

Graph coloring · qf-mod31-coloring-relabel-invariance · candidate





## 83. Injective Renaming of Used Colors Preserves the Vertex Partition

Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5]]. Normalize each edge increasingly, then sort edges lexicographically; reversed duplicates and loops are forbidden. Keep isolated vertices in the carrier. No directed-graph conclusion is inferred. Total vertex colors=[0, 1, 0, 1, 0, 1] are labels, not cardinalities. Only adjacent endpoints must differ; nonadjacent vertices may share a color. A proper k-coloring uses at most k distinct labels, even if their numerical names exceed k−1. One such coloring gives only a chromatic-number upper bound, not optimality or a general greedy guarantee. Color relabel table=[17, 47] is indexed by the original color labels and must be injective on the labels actually used. Give (colors,recolored,proper_before?,proper_after?,same_conflicts?,color_names_do_not_change_vertex_partition).

View answer and explanation([0, 1, 0, 1, 0, 1], [17, 47, 17, 47, 17, 47], Yes, Yes, Yes, color_names_do_not_change_vertex_partition)

Apply the table to every vertex in original order. Distinct used colors must remain distinct, so exactly the same edges remain monochromatic. Even the conflicts of a failed coloring are preserved, not repaired. The exact result, retaining the requested order, is ([0, 1, 0, 1, 0, 1], [17, 47, 17, 47, 17, 47], Yes, Yes, Yes, color_names_do_not_change_vertex_partition).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r05-graph-coloring-0083/)

Graph coloring · qf-mod31-coloring-relabel-invariance · candidate





## 84. Injective Renaming of Used Colors Preserves the Vertex Partition

Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5]]. Normalize each edge increasingly, then sort edges lexicographically; reversed duplicates and loops are forbidden. Keep isolated vertices in the carrier. No directed-graph conclusion is inferred. Total vertex colors=[0, 1, 0, 1, 0, 1] are labels, not cardinalities. Only adjacent endpoints must differ; nonadjacent vertices may share a color. A proper k-coloring uses at most k distinct labels, even if their numerical names exceed k−1. One such coloring gives only a chromatic-number upper bound, not optimality or a general greedy guarantee. Color relabel table=[18, 48] is indexed by the original color labels and must be injective on the labels actually used. Give (colors,recolored,proper_before?,proper_after?,same_conflicts?,color_names_do_not_change_vertex_partition).

View answer and explanation([0, 1, 0, 1, 0, 1], [18, 48, 18, 48, 18, 48], Yes, Yes, Yes, color_names_do_not_change_vertex_partition)

Apply the table to every vertex in original order. Distinct used colors must remain distinct, so exactly the same edges remain monochromatic. Even the conflicts of a failed coloring are preserved, not repaired. The exact result, retaining the requested order, is ([0, 1, 0, 1, 0, 1], [18, 48, 18, 48, 18, 48], Yes, Yes, Yes, color_names_do_not_change_vertex_partition).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r05-graph-coloring-0084/)

Graph coloring · qf-mod31-coloring-relabel-invariance · candidate





## 85. Injective Renaming of Used Colors Preserves the Vertex Partition

Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5]]. Normalize each edge increasingly, then sort edges lexicographically; reversed duplicates and loops are forbidden. Keep isolated vertices in the carrier. No directed-graph conclusion is inferred. Total vertex colors=[0, 1, 0, 1, 0, 1] are labels, not cardinalities. Only adjacent endpoints must differ; nonadjacent vertices may share a color. A proper k-coloring uses at most k distinct labels, even if their numerical names exceed k−1. One such coloring gives only a chromatic-number upper bound, not optimality or a general greedy guarantee. Color relabel table=[19, 49] is indexed by the original color labels and must be injective on the labels actually used. Give (colors,recolored,proper_before?,proper_after?,same_conflicts?,color_names_do_not_change_vertex_partition).

View answer and explanation([0, 1, 0, 1, 0, 1], [19, 49, 19, 49, 19, 49], Yes, Yes, Yes, color_names_do_not_change_vertex_partition)

Apply the table to every vertex in original order. Distinct used colors must remain distinct, so exactly the same edges remain monochromatic. Even the conflicts of a failed coloring are preserved, not repaired. The exact result, retaining the requested order, is ([0, 1, 0, 1, 0, 1], [19, 49, 19, 49, 19, 49], Yes, Yes, Yes, color_names_do_not_change_vertex_partition).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r05-graph-coloring-0085/)

Graph coloring · qf-mod31-coloring-relabel-invariance · candidate





## 86. Injective Renaming of Used Colors Preserves the Vertex Partition

Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5]]. Normalize each edge increasingly, then sort edges lexicographically; reversed duplicates and loops are forbidden. Keep isolated vertices in the carrier. No directed-graph conclusion is inferred. Total vertex colors=[0, 1, 0, 1, 0, 1] are labels, not cardinalities. Only adjacent endpoints must differ; nonadjacent vertices may share a color. A proper k-coloring uses at most k distinct labels, even if their numerical names exceed k−1. One such coloring gives only a chromatic-number upper bound, not optimality or a general greedy guarantee. Color relabel table=[20, 50] is indexed by the original color labels and must be injective on the labels actually used. Give (colors,recolored,proper_before?,proper_after?,same_conflicts?,color_names_do_not_change_vertex_partition).

View answer and explanation([0, 1, 0, 1, 0, 1], [20, 50, 20, 50, 20, 50], Yes, Yes, Yes, color_names_do_not_change_vertex_partition)

Apply the table to every vertex in original order. Distinct used colors must remain distinct, so exactly the same edges remain monochromatic. Even the conflicts of a failed coloring are preserved, not repaired. The exact result, retaining the requested order, is ([0, 1, 0, 1, 0, 1], [20, 50, 20, 50, 20, 50], Yes, Yes, Yes, color_names_do_not_change_vertex_partition).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r05-graph-coloring-0086/)

Graph coloring · qf-mod31-coloring-relabel-invariance · candidate





## 87. Injective Renaming of Used Colors Preserves the Vertex Partition

Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5]]. Normalize each edge increasingly, then sort edges lexicographically; reversed duplicates and loops are forbidden. Keep isolated vertices in the carrier. No directed-graph conclusion is inferred. Total vertex colors=[0, 1, 0, 1, 0, 1] are labels, not cardinalities. Only adjacent endpoints must differ; nonadjacent vertices may share a color. A proper k-coloring uses at most k distinct labels, even if their numerical names exceed k−1. One such coloring gives only a chromatic-number upper bound, not optimality or a general greedy guarantee. Color relabel table=[21, 51] is indexed by the original color labels and must be injective on the labels actually used. Give (colors,recolored,proper_before?,proper_after?,same_conflicts?,color_names_do_not_change_vertex_partition).

View answer and explanation([0, 1, 0, 1, 0, 1], [21, 51, 21, 51, 21, 51], Yes, Yes, Yes, color_names_do_not_change_vertex_partition)

Apply the table to every vertex in original order. Distinct used colors must remain distinct, so exactly the same edges remain monochromatic. Even the conflicts of a failed coloring are preserved, not repaired. The exact result, retaining the requested order, is ([0, 1, 0, 1, 0, 1], [21, 51, 21, 51, 21, 51], Yes, Yes, Yes, color_names_do_not_change_vertex_partition).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r05-graph-coloring-0087/)

Graph coloring · qf-mod31-coloring-relabel-invariance · candidate





## 88. Injective Renaming of Used Colors Preserves the Vertex Partition

Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5]]. Normalize each edge increasingly, then sort edges lexicographically; reversed duplicates and loops are forbidden. Keep isolated vertices in the carrier. No directed-graph conclusion is inferred. Total vertex colors=[0, 1, 0, 1, 0, 1] are labels, not cardinalities. Only adjacent endpoints must differ; nonadjacent vertices may share a color. A proper k-coloring uses at most k distinct labels, even if their numerical names exceed k−1. One such coloring gives only a chromatic-number upper bound, not optimality or a general greedy guarantee. Color relabel table=[22, 52] is indexed by the original color labels and must be injective on the labels actually used. Give (colors,recolored,proper_before?,proper_after?,same_conflicts?,color_names_do_not_change_vertex_partition).

View answer and explanation([0, 1, 0, 1, 0, 1], [22, 52, 22, 52, 22, 52], Yes, Yes, Yes, color_names_do_not_change_vertex_partition)

Apply the table to every vertex in original order. Distinct used colors must remain distinct, so exactly the same edges remain monochromatic. Even the conflicts of a failed coloring are preserved, not repaired. The exact result, retaining the requested order, is ([0, 1, 0, 1, 0, 1], [22, 52, 22, 52, 22, 52], Yes, Yes, Yes, color_names_do_not_change_vertex_partition).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r05-graph-coloring-0088/)

Graph coloring · qf-mod31-coloring-relabel-invariance · candidate





## 89. Injective Renaming of Used Colors Preserves the Vertex Partition

Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5]]. Normalize each edge increasingly, then sort edges lexicographically; reversed duplicates and loops are forbidden. Keep isolated vertices in the carrier. No directed-graph conclusion is inferred. Total vertex colors=[0, 1, 0, 1, 0, 1] are labels, not cardinalities. Only adjacent endpoints must differ; nonadjacent vertices may share a color. A proper k-coloring uses at most k distinct labels, even if their numerical names exceed k−1. One such coloring gives only a chromatic-number upper bound, not optimality or a general greedy guarantee. Color relabel table=[23, 53] is indexed by the original color labels and must be injective on the labels actually used. Give (colors,recolored,proper_before?,proper_after?,same_conflicts?,color_names_do_not_change_vertex_partition).

View answer and explanation([0, 1, 0, 1, 0, 1], [23, 53, 23, 53, 23, 53], Yes, Yes, Yes, color_names_do_not_change_vertex_partition)

Apply the table to every vertex in original order. Distinct used colors must remain distinct, so exactly the same edges remain monochromatic. Even the conflicts of a failed coloring are preserved, not repaired. The exact result, retaining the requested order, is ([0, 1, 0, 1, 0, 1], [23, 53, 23, 53, 23, 53], Yes, Yes, Yes, color_names_do_not_change_vertex_partition).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r05-graph-coloring-0089/)

Graph coloring · qf-mod31-coloring-relabel-invariance · candidate





## 90. Injective Renaming of Used Colors Preserves the Vertex Partition

Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5]]. Normalize each edge increasingly, then sort edges lexicographically; reversed duplicates and loops are forbidden. Keep isolated vertices in the carrier. No directed-graph conclusion is inferred. Total vertex colors=[0, 1, 0, 1, 0, 1] are labels, not cardinalities. Only adjacent endpoints must differ; nonadjacent vertices may share a color. A proper k-coloring uses at most k distinct labels, even if their numerical names exceed k−1. One such coloring gives only a chromatic-number upper bound, not optimality or a general greedy guarantee. Color relabel table=[24, 54] is indexed by the original color labels and must be injective on the labels actually used. Give (colors,recolored,proper_before?,proper_after?,same_conflicts?,color_names_do_not_change_vertex_partition).

View answer and explanation([0, 1, 0, 1, 0, 1], [24, 54, 24, 54, 24, 54], Yes, Yes, Yes, color_names_do_not_change_vertex_partition)

Apply the table to every vertex in original order. Distinct used colors must remain distinct, so exactly the same edges remain monochromatic. Even the conflicts of a failed coloring are preserved, not repaired. The exact result, retaining the requested order, is ([0, 1, 0, 1, 0, 1], [24, 54, 24, 54, 24, 54], Yes, Yes, Yes, color_names_do_not_change_vertex_partition).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r05-graph-coloring-0090/)

Graph coloring · qf-mod31-coloring-relabel-invariance · candidate





## 91. Injective Renaming of Used Colors Preserves the Vertex Partition

Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5]]. Normalize each edge increasingly, then sort edges lexicographically; reversed duplicates and loops are forbidden. Keep isolated vertices in the carrier. No directed-graph conclusion is inferred. Total vertex colors=[0, 1, 0, 1, 0, 1] are labels, not cardinalities. Only adjacent endpoints must differ; nonadjacent vertices may share a color. A proper k-coloring uses at most k distinct labels, even if their numerical names exceed k−1. One such coloring gives only a chromatic-number upper bound, not optimality or a general greedy guarantee. Color relabel table=[25, 55] is indexed by the original color labels and must be injective on the labels actually used. Give (colors,recolored,proper_before?,proper_after?,same_conflicts?,color_names_do_not_change_vertex_partition).

View answer and explanation([0, 1, 0, 1, 0, 1], [25, 55, 25, 55, 25, 55], Yes, Yes, Yes, color_names_do_not_change_vertex_partition)

Apply the table to every vertex in original order. Distinct used colors must remain distinct, so exactly the same edges remain monochromatic. Even the conflicts of a failed coloring are preserved, not repaired. The exact result, retaining the requested order, is ([0, 1, 0, 1, 0, 1], [25, 55, 25, 55, 25, 55], Yes, Yes, Yes, color_names_do_not_change_vertex_partition).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r05-graph-coloring-0091/)

Graph coloring · qf-mod31-coloring-relabel-invariance · candidate





## 92. Injective Renaming of Used Colors Preserves the Vertex Partition

Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5]]. Normalize each edge increasingly, then sort edges lexicographically; reversed duplicates and loops are forbidden. Keep isolated vertices in the carrier. No directed-graph conclusion is inferred. Total vertex colors=[0, 1, 0, 1, 0, 1] are labels, not cardinalities. Only adjacent endpoints must differ; nonadjacent vertices may share a color. A proper k-coloring uses at most k distinct labels, even if their numerical names exceed k−1. One such coloring gives only a chromatic-number upper bound, not optimality or a general greedy guarantee. Color relabel table=[26, 56] is indexed by the original color labels and must be injective on the labels actually used. Give (colors,recolored,proper_before?,proper_after?,same_conflicts?,color_names_do_not_change_vertex_partition).

View answer and explanation([0, 1, 0, 1, 0, 1], [26, 56, 26, 56, 26, 56], Yes, Yes, Yes, color_names_do_not_change_vertex_partition)

Apply the table to every vertex in original order. Distinct used colors must remain distinct, so exactly the same edges remain monochromatic. Even the conflicts of a failed coloring are preserved, not repaired. The exact result, retaining the requested order, is ([0, 1, 0, 1, 0, 1], [26, 56, 26, 56, 26, 56], Yes, Yes, Yes, color_names_do_not_change_vertex_partition).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r05-graph-coloring-0092/)

Graph coloring · qf-mod31-coloring-relabel-invariance · candidate





## 93. Injective Renaming of Used Colors Preserves the Vertex Partition

Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5]]. Normalize each edge increasingly, then sort edges lexicographically; reversed duplicates and loops are forbidden. Keep isolated vertices in the carrier. No directed-graph conclusion is inferred. Total vertex colors=[0, 1, 0, 1, 0, 1] are labels, not cardinalities. Only adjacent endpoints must differ; nonadjacent vertices may share a color. A proper k-coloring uses at most k distinct labels, even if their numerical names exceed k−1. One such coloring gives only a chromatic-number upper bound, not optimality or a general greedy guarantee. Color relabel table=[27, 57] is indexed by the original color labels and must be injective on the labels actually used. Give (colors,recolored,proper_before?,proper_after?,same_conflicts?,color_names_do_not_change_vertex_partition).

View answer and explanation([0, 1, 0, 1, 0, 1], [27, 57, 27, 57, 27, 57], Yes, Yes, Yes, color_names_do_not_change_vertex_partition)

Apply the table to every vertex in original order. Distinct used colors must remain distinct, so exactly the same edges remain monochromatic. Even the conflicts of a failed coloring are preserved, not repaired. The exact result, retaining the requested order, is ([0, 1, 0, 1, 0, 1], [27, 57, 27, 57, 27, 57], Yes, Yes, Yes, color_names_do_not_change_vertex_partition).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r05-graph-coloring-0093/)

Graph coloring · qf-mod31-coloring-relabel-invariance · candidate





## 94. Injective Renaming of Used Colors Preserves the Vertex Partition

Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5]]. Normalize each edge increasingly, then sort edges lexicographically; reversed duplicates and loops are forbidden. Keep isolated vertices in the carrier. No directed-graph conclusion is inferred. Total vertex colors=[0, 1, 0, 1, 0, 1] are labels, not cardinalities. Only adjacent endpoints must differ; nonadjacent vertices may share a color. A proper k-coloring uses at most k distinct labels, even if their numerical names exceed k−1. One such coloring gives only a chromatic-number upper bound, not optimality or a general greedy guarantee. Color relabel table=[28, 58] is indexed by the original color labels and must be injective on the labels actually used. Give (colors,recolored,proper_before?,proper_after?,same_conflicts?,color_names_do_not_change_vertex_partition).

View answer and explanation([0, 1, 0, 1, 0, 1], [28, 58, 28, 58, 28, 58], Yes, Yes, Yes, color_names_do_not_change_vertex_partition)

Apply the table to every vertex in original order. Distinct used colors must remain distinct, so exactly the same edges remain monochromatic. Even the conflicts of a failed coloring are preserved, not repaired. The exact result, retaining the requested order, is ([0, 1, 0, 1, 0, 1], [28, 58, 28, 58, 28, 58], Yes, Yes, Yes, color_names_do_not_change_vertex_partition).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r05-graph-coloring-0094/)

Graph coloring · qf-mod31-coloring-relabel-invariance · candidate





## 95. Injective Renaming of Used Colors Preserves the Vertex Partition

Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5]]. Normalize each edge increasingly, then sort edges lexicographically; reversed duplicates and loops are forbidden. Keep isolated vertices in the carrier. No directed-graph conclusion is inferred. Total vertex colors=[0, 1, 0, 1, 0, 1] are labels, not cardinalities. Only adjacent endpoints must differ; nonadjacent vertices may share a color. A proper k-coloring uses at most k distinct labels, even if their numerical names exceed k−1. One such coloring gives only a chromatic-number upper bound, not optimality or a general greedy guarantee. Color relabel table=[29, 59] is indexed by the original color labels and must be injective on the labels actually used. Give (colors,recolored,proper_before?,proper_after?,same_conflicts?,color_names_do_not_change_vertex_partition).

View answer and explanation([0, 1, 0, 1, 0, 1], [29, 59, 29, 59, 29, 59], Yes, Yes, Yes, color_names_do_not_change_vertex_partition)

Apply the table to every vertex in original order. Distinct used colors must remain distinct, so exactly the same edges remain monochromatic. Even the conflicts of a failed coloring are preserved, not repaired. The exact result, retaining the requested order, is ([0, 1, 0, 1, 0, 1], [29, 59, 29, 59, 29, 59], Yes, Yes, Yes, color_names_do_not_change_vertex_partition).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r05-graph-coloring-0095/)

Graph coloring · qf-mod31-coloring-relabel-invariance · candidate





## 96. Injective Renaming of Used Colors Preserves the Vertex Partition

Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5]]. Normalize each edge increasingly, then sort edges lexicographically; reversed duplicates and loops are forbidden. Keep isolated vertices in the carrier. No directed-graph conclusion is inferred. Total vertex colors=[0, 1, 0, 1, 0, 1] are labels, not cardinalities. Only adjacent endpoints must differ; nonadjacent vertices may share a color. A proper k-coloring uses at most k distinct labels, even if their numerical names exceed k−1. One such coloring gives only a chromatic-number upper bound, not optimality or a general greedy guarantee. Color relabel table=[30, 60] is indexed by the original color labels and must be injective on the labels actually used. Give (colors,recolored,proper_before?,proper_after?,same_conflicts?,color_names_do_not_change_vertex_partition).

View answer and explanation([0, 1, 0, 1, 0, 1], [30, 60, 30, 60, 30, 60], Yes, Yes, Yes, color_names_do_not_change_vertex_partition)

Apply the table to every vertex in original order. Distinct used colors must remain distinct, so exactly the same edges remain monochromatic. Even the conflicts of a failed coloring are preserved, not repaired. The exact result, retaining the requested order, is ([0, 1, 0, 1, 0, 1], [30, 60, 30, 60, 30, 60], Yes, Yes, Yes, color_names_do_not_change_vertex_partition).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r05-graph-coloring-0096/)

Graph coloring · qf-mod31-coloring-relabel-invariance · candidate





## 97. Injective Renaming of Used Colors Preserves the Vertex Partition

Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5]]. Normalize each edge increasingly, then sort edges lexicographically; reversed duplicates and loops are forbidden. Keep isolated vertices in the carrier. No directed-graph conclusion is inferred. Total vertex colors=[0, 1, 0, 1, 0, 1] are labels, not cardinalities. Only adjacent endpoints must differ; nonadjacent vertices may share a color. A proper k-coloring uses at most k distinct labels, even if their numerical names exceed k−1. One such coloring gives only a chromatic-number upper bound, not optimality or a general greedy guarantee. Color relabel table=[31, 61] is indexed by the original color labels and must be injective on the labels actually used. Give (colors,recolored,proper_before?,proper_after?,same_conflicts?,color_names_do_not_change_vertex_partition).

View answer and explanation([0, 1, 0, 1, 0, 1], [31, 61, 31, 61, 31, 61], Yes, Yes, Yes, color_names_do_not_change_vertex_partition)

Apply the table to every vertex in original order. Distinct used colors must remain distinct, so exactly the same edges remain monochromatic. Even the conflicts of a failed coloring are preserved, not repaired. The exact result, retaining the requested order, is ([0, 1, 0, 1, 0, 1], [31, 61, 31, 61, 31, 61], Yes, Yes, Yes, color_names_do_not_change_vertex_partition).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r05-graph-coloring-0097/)

Graph coloring · qf-mod31-coloring-relabel-invariance · candidate





## 98. Injective Renaming of Used Colors Preserves the Vertex Partition

Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5]]. Normalize each edge increasingly, then sort edges lexicographically; reversed duplicates and loops are forbidden. Keep isolated vertices in the carrier. No directed-graph conclusion is inferred. Total vertex colors=[0, 1, 0, 1, 0, 1] are labels, not cardinalities. Only adjacent endpoints must differ; nonadjacent vertices may share a color. A proper k-coloring uses at most k distinct labels, even if their numerical names exceed k−1. One such coloring gives only a chromatic-number upper bound, not optimality or a general greedy guarantee. Color relabel table=[32, 62] is indexed by the original color labels and must be injective on the labels actually used. Give (colors,recolored,proper_before?,proper_after?,same_conflicts?,color_names_do_not_change_vertex_partition).

View answer and explanation([0, 1, 0, 1, 0, 1], [32, 62, 32, 62, 32, 62], Yes, Yes, Yes, color_names_do_not_change_vertex_partition)

Apply the table to every vertex in original order. Distinct used colors must remain distinct, so exactly the same edges remain monochromatic. Even the conflicts of a failed coloring are preserved, not repaired. The exact result, retaining the requested order, is ([0, 1, 0, 1, 0, 1], [32, 62, 32, 62, 32, 62], Yes, Yes, Yes, color_names_do_not_change_vertex_partition).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r05-graph-coloring-0098/)

Graph coloring · qf-mod31-coloring-relabel-invariance · candidate





## 99. Injective Renaming of Used Colors Preserves the Vertex Partition

Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5]]. Normalize each edge increasingly, then sort edges lexicographically; reversed duplicates and loops are forbidden. Keep isolated vertices in the carrier. No directed-graph conclusion is inferred. Total vertex colors=[0, 1, 0, 1, 0, 1] are labels, not cardinalities. Only adjacent endpoints must differ; nonadjacent vertices may share a color. A proper k-coloring uses at most k distinct labels, even if their numerical names exceed k−1. One such coloring gives only a chromatic-number upper bound, not optimality or a general greedy guarantee. Color relabel table=[33, 63] is indexed by the original color labels and must be injective on the labels actually used. Give (colors,recolored,proper_before?,proper_after?,same_conflicts?,color_names_do_not_change_vertex_partition).

View answer and explanation([0, 1, 0, 1, 0, 1], [33, 63, 33, 63, 33, 63], Yes, Yes, Yes, color_names_do_not_change_vertex_partition)

Apply the table to every vertex in original order. Distinct used colors must remain distinct, so exactly the same edges remain monochromatic. Even the conflicts of a failed coloring are preserved, not repaired. The exact result, retaining the requested order, is ([0, 1, 0, 1, 0, 1], [33, 63, 33, 63, 33, 63], Yes, Yes, Yes, color_names_do_not_change_vertex_partition).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r05-graph-coloring-0099/)

Graph coloring · qf-mod31-coloring-relabel-invariance · candidate





## 100. Injective Renaming of Used Colors Preserves the Vertex Partition

Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5]]. Normalize each edge increasingly, then sort edges lexicographically; reversed duplicates and loops are forbidden. Keep isolated vertices in the carrier. No directed-graph conclusion is inferred. Total vertex colors=[0, 1, 0, 1, 0, 1] are labels, not cardinalities. Only adjacent endpoints must differ; nonadjacent vertices may share a color. A proper k-coloring uses at most k distinct labels, even if their numerical names exceed k−1. One such coloring gives only a chromatic-number upper bound, not optimality or a general greedy guarantee. Color relabel table=[34, 64] is indexed by the original color labels and must be injective on the labels actually used. Give (colors,recolored,proper_before?,proper_after?,same_conflicts?,color_names_do_not_change_vertex_partition).

View answer and explanation([0, 1, 0, 1, 0, 1], [34, 64, 34, 64, 34, 64], Yes, Yes, Yes, color_names_do_not_change_vertex_partition)

Apply the table to every vertex in original order. Distinct used colors must remain distinct, so exactly the same edges remain monochromatic. Even the conflicts of a failed coloring are preserved, not repaired. The exact result, retaining the requested order, is ([0, 1, 0, 1, 0, 1], [34, 64, 34, 64, 34, 64], Yes, Yes, Yes, color_names_do_not_change_vertex_partition).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r05-graph-coloring-0100/)

Graph coloring · qf-mod31-coloring-relabel-invariance · candidate



[Canonical question JSONL](https://ocme.evemisslab.com/data/questions/batches/batch-20261014-advance-20261031-r05-graph-coloring-foundations/questions.jsonl) · [English JSONL](https://ocme.evemisslab.com/data/i18n/questions/batch-20261014-advance-20261031-r05-graph-coloring-foundations.en.jsonl) · [Validation report](https://ocme.evemisslab.com/data/questions/batches/batch-20261014-advance-20261031-r05-graph-coloring-foundations/validation-report.json)