Complete Coloring Audits, Nonoptimal Upper Bounds and Color-Name Invariance
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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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
Graph coloring · qf-mod31-coloring-relabel-invariance · candidate
Canonical question JSONL · English JSONL · Validation report