# Proper Palettes, Exact Chromatic Minima and Independent Certificate Flags

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

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

batch-20261014-advance-20261106-r01-chromatic-number-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.

Field order uses a source-bound presentation supplement; canonical JSONL and mathematical answers are unchanged.

[Field-order and source-binding data](https://ocme.evemisslab.com/data/i18n/question-output-protocols.json)



## 1. Proper Coloring Alone Does Not Prove the Chromatic Minimum

Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 5], [1, 2], [2, 3], [3, 4], [4, 5]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Supplied colors=[0, 1, 0, 1, 0, 1] in vertex order. Color labels are nonnegative integer palette indices; retain their actual labels. Properness checks every edge, and using palette [0,k) is separate from merely using at most k distinct labels. Give (colors, used_colors, color_count, proper, boundary).

View answer and explanation([0, 1, 0, 1, 0, 1], [0, 1], 2, Yes, proper_coloring_only)

Report the original color array, increasingly ordered used labels, their count and all-edge properness. A valid coloring supplies an upper bound, not automatically equality with the chromatic number. The exact result, retaining the requested order, is ([0, 1, 0, 1, 0, 1], [0, 1], 2, Yes, proper_coloring_only).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r01-chromatic-number-0001/)

Chromatic number · qf-mod37-chromatic-proper-audit · candidate





## 2. Proper Coloring Alone Does Not Prove the Chromatic Minimum

Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 5], [1, 2], [2, 3], [3, 4], [4, 5]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Supplied colors=[0, 1, 2, 0, 1, 2] in vertex order. Color labels are nonnegative integer palette indices; retain their actual labels. Properness checks every edge, and using palette [0,k) is separate from merely using at most k distinct labels. Give (colors, used_colors, color_count, proper, boundary).

View answer and explanation([0, 1, 2, 0, 1, 2], [0, 1, 2], 3, Yes, proper_coloring_only)

Report the original color array, increasingly ordered used labels, their count and all-edge properness. A valid coloring supplies an upper bound, not automatically equality with the chromatic number. The exact result, retaining the requested order, is ([0, 1, 2, 0, 1, 2], [0, 1, 2], 3, Yes, proper_coloring_only).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r01-chromatic-number-0002/)

Chromatic number · qf-mod37-chromatic-proper-audit · candidate





## 3. Proper Coloring Alone Does Not Prove the Chromatic Minimum

Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 3], [0, 5], [1, 2], [2, 3], [3, 4], [4, 5]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Supplied colors=[0, 1, 0, 1, 0, 1] in vertex order. Color labels are nonnegative integer palette indices; retain their actual labels. Properness checks every edge, and using palette [0,k) is separate from merely using at most k distinct labels. Give (colors, used_colors, color_count, proper, boundary).

View answer and explanation([0, 1, 0, 1, 0, 1], [0, 1], 2, Yes, proper_coloring_only)

Report the original color array, increasingly ordered used labels, their count and all-edge properness. A valid coloring supplies an upper bound, not automatically equality with the chromatic number. The exact result, retaining the requested order, is ([0, 1, 0, 1, 0, 1], [0, 1], 2, Yes, proper_coloring_only).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r01-chromatic-number-0003/)

Chromatic number · qf-mod37-chromatic-proper-audit · candidate





## 4. Proper Coloring Alone Does Not Prove the Chromatic Minimum

Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 3], [0, 5], [1, 2], [2, 3], [3, 4], [4, 5]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Supplied colors=[0, 1, 2, 1, 0, 1] in vertex order. Color labels are nonnegative integer palette indices; retain their actual labels. Properness checks every edge, and using palette [0,k) is separate from merely using at most k distinct labels. Give (colors, used_colors, color_count, proper, boundary).

View answer and explanation([0, 1, 2, 1, 0, 1], [0, 1, 2], 3, Yes, proper_coloring_only)

Report the original color array, increasingly ordered used labels, their count and all-edge properness. A valid coloring supplies an upper bound, not automatically equality with the chromatic number. The exact result, retaining the requested order, is ([0, 1, 2, 1, 0, 1], [0, 1, 2], 3, Yes, proper_coloring_only).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r01-chromatic-number-0004/)

Chromatic number · qf-mod37-chromatic-proper-audit · candidate





## 5. Proper Coloring Alone Does Not Prove the Chromatic Minimum

Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 5], [1, 2], [1, 3], [2, 3], [3, 4], [4, 5]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Supplied colors=[0, 1, 0, 2, 0, 1] in vertex order. Color labels are nonnegative integer palette indices; retain their actual labels. Properness checks every edge, and using palette [0,k) is separate from merely using at most k distinct labels. Give (colors, used_colors, color_count, proper, boundary).

View answer and explanation([0, 1, 0, 2, 0, 1], [0, 1, 2], 3, Yes, proper_coloring_only)

Report the original color array, increasingly ordered used labels, their count and all-edge properness. A valid coloring supplies an upper bound, not automatically equality with the chromatic number. The exact result, retaining the requested order, is ([0, 1, 0, 2, 0, 1], [0, 1, 2], 3, Yes, proper_coloring_only).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r01-chromatic-number-0005/)

Chromatic number · qf-mod37-chromatic-proper-audit · candidate





## 6. Proper Coloring Alone Does Not Prove the Chromatic Minimum

Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 5], [1, 2], [1, 3], [2, 3], [3, 4], [4, 5]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Supplied colors=[0, 1, 2, 0, 1, 2] in vertex order. Color labels are nonnegative integer palette indices; retain their actual labels. Properness checks every edge, and using palette [0,k) is separate from merely using at most k distinct labels. Give (colors, used_colors, color_count, proper, boundary).

View answer and explanation([0, 1, 2, 0, 1, 2], [0, 1, 2], 3, Yes, proper_coloring_only)

Report the original color array, increasingly ordered used labels, their count and all-edge properness. A valid coloring supplies an upper bound, not automatically equality with the chromatic number. The exact result, retaining the requested order, is ([0, 1, 2, 0, 1, 2], [0, 1, 2], 3, Yes, proper_coloring_only).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r01-chromatic-number-0006/)

Chromatic number · qf-mod37-chromatic-proper-audit · candidate





## 7. Proper Coloring Alone Does Not Prove the Chromatic Minimum

Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 3], [0, 5], [1, 2], [1, 3], [2, 3], [3, 4], [4, 5]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Supplied colors=[0, 1, 0, 2, 0, 1] in vertex order. Color labels are nonnegative integer palette indices; retain their actual labels. Properness checks every edge, and using palette [0,k) is separate from merely using at most k distinct labels. Give (colors, used_colors, color_count, proper, boundary).

View answer and explanation([0, 1, 0, 2, 0, 1], [0, 1, 2], 3, Yes, proper_coloring_only)

Report the original color array, increasingly ordered used labels, their count and all-edge properness. A valid coloring supplies an upper bound, not automatically equality with the chromatic number. The exact result, retaining the requested order, is ([0, 1, 0, 2, 0, 1], [0, 1, 2], 3, Yes, proper_coloring_only).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r01-chromatic-number-0007/)

Chromatic number · qf-mod37-chromatic-proper-audit · candidate





## 8. Proper Coloring Alone Does Not Prove the Chromatic Minimum

Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 3], [0, 5], [1, 2], [1, 3], [2, 3], [3, 4], [4, 5]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Supplied colors=[0, 1, 2, 3, 0, 1] in vertex order. Color labels are nonnegative integer palette indices; retain their actual labels. Properness checks every edge, and using palette [0,k) is separate from merely using at most k distinct labels. Give (colors, used_colors, color_count, proper, boundary).

View answer and explanation([0, 1, 2, 3, 0, 1], [0, 1, 2, 3], 4, Yes, proper_coloring_only)

Report the original color array, increasingly ordered used labels, their count and all-edge properness. A valid coloring supplies an upper bound, not automatically equality with the chromatic number. The exact result, retaining the requested order, is ([0, 1, 2, 3, 0, 1], [0, 1, 2, 3], 4, Yes, proper_coloring_only).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r01-chromatic-number-0008/)

Chromatic number · qf-mod37-chromatic-proper-audit · candidate





## 9. Proper Coloring Alone Does Not Prove the Chromatic Minimum

Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 5], [1, 2], [1, 4], [2, 3], [3, 4], [4, 5]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Supplied colors=[0, 1, 0, 1, 0, 1] in vertex order. Color labels are nonnegative integer palette indices; retain their actual labels. Properness checks every edge, and using palette [0,k) is separate from merely using at most k distinct labels. Give (colors, used_colors, color_count, proper, boundary).

View answer and explanation([0, 1, 0, 1, 0, 1], [0, 1], 2, Yes, proper_coloring_only)

Report the original color array, increasingly ordered used labels, their count and all-edge properness. A valid coloring supplies an upper bound, not automatically equality with the chromatic number. The exact result, retaining the requested order, is ([0, 1, 0, 1, 0, 1], [0, 1], 2, Yes, proper_coloring_only).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r01-chromatic-number-0009/)

Chromatic number · qf-mod37-chromatic-proper-audit · candidate





## 10. Proper Coloring Alone Does Not Prove the Chromatic Minimum

Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 5], [1, 2], [1, 4], [2, 3], [3, 4], [4, 5]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Supplied colors=[0, 1, 2, 0, 2, 1] in vertex order. Color labels are nonnegative integer palette indices; retain their actual labels. Properness checks every edge, and using palette [0,k) is separate from merely using at most k distinct labels. Give (colors, used_colors, color_count, proper, boundary).

View answer and explanation([0, 1, 2, 0, 2, 1], [0, 1, 2], 3, Yes, proper_coloring_only)

Report the original color array, increasingly ordered used labels, their count and all-edge properness. A valid coloring supplies an upper bound, not automatically equality with the chromatic number. The exact result, retaining the requested order, is ([0, 1, 2, 0, 2, 1], [0, 1, 2], 3, Yes, proper_coloring_only).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r01-chromatic-number-0010/)

Chromatic number · qf-mod37-chromatic-proper-audit · candidate





## 11. Proper Coloring Alone Does Not Prove the Chromatic Minimum

Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 3], [0, 5], [1, 2], [1, 4], [2, 3], [3, 4], [4, 5]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Supplied colors=[0, 1, 0, 1, 0, 1] in vertex order. Color labels are nonnegative integer palette indices; retain their actual labels. Properness checks every edge, and using palette [0,k) is separate from merely using at most k distinct labels. Give (colors, used_colors, color_count, proper, boundary).

View answer and explanation([0, 1, 0, 1, 0, 1], [0, 1], 2, Yes, proper_coloring_only)

Report the original color array, increasingly ordered used labels, their count and all-edge properness. A valid coloring supplies an upper bound, not automatically equality with the chromatic number. The exact result, retaining the requested order, is ([0, 1, 0, 1, 0, 1], [0, 1], 2, Yes, proper_coloring_only).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r01-chromatic-number-0011/)

Chromatic number · qf-mod37-chromatic-proper-audit · candidate





## 12. Proper Coloring Alone Does Not Prove the Chromatic Minimum

Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 3], [0, 5], [1, 2], [1, 4], [2, 3], [3, 4], [4, 5]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Supplied colors=[0, 1, 2, 1, 0, 1] in vertex order. Color labels are nonnegative integer palette indices; retain their actual labels. Properness checks every edge, and using palette [0,k) is separate from merely using at most k distinct labels. Give (colors, used_colors, color_count, proper, boundary).

View answer and explanation([0, 1, 2, 1, 0, 1], [0, 1, 2], 3, Yes, proper_coloring_only)

Report the original color array, increasingly ordered used labels, their count and all-edge properness. A valid coloring supplies an upper bound, not automatically equality with the chromatic number. The exact result, retaining the requested order, is ([0, 1, 2, 1, 0, 1], [0, 1, 2], 3, Yes, proper_coloring_only).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r01-chromatic-number-0012/)

Chromatic number · qf-mod37-chromatic-proper-audit · candidate





## 13. Proper Coloring Alone Does Not Prove the Chromatic Minimum

Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 5], [1, 2], [1, 3], [1, 4], [2, 3], [3, 4], [4, 5]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Supplied colors=[0, 1, 0, 2, 0, 1] in vertex order. Color labels are nonnegative integer palette indices; retain their actual labels. Properness checks every edge, and using palette [0,k) is separate from merely using at most k distinct labels. Give (colors, used_colors, color_count, proper, boundary).

View answer and explanation([0, 1, 0, 2, 0, 1], [0, 1, 2], 3, Yes, proper_coloring_only)

Report the original color array, increasingly ordered used labels, their count and all-edge properness. A valid coloring supplies an upper bound, not automatically equality with the chromatic number. The exact result, retaining the requested order, is ([0, 1, 0, 2, 0, 1], [0, 1, 2], 3, Yes, proper_coloring_only).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r01-chromatic-number-0013/)

Chromatic number · qf-mod37-chromatic-proper-audit · candidate





## 14. Proper Coloring Alone Does Not Prove the Chromatic Minimum

Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 5], [1, 2], [1, 3], [1, 4], [2, 3], [3, 4], [4, 5]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Supplied colors=[0, 1, 2, 0, 2, 1] in vertex order. Color labels are nonnegative integer palette indices; retain their actual labels. Properness checks every edge, and using palette [0,k) is separate from merely using at most k distinct labels. Give (colors, used_colors, color_count, proper, boundary).

View answer and explanation([0, 1, 2, 0, 2, 1], [0, 1, 2], 3, Yes, proper_coloring_only)

Report the original color array, increasingly ordered used labels, their count and all-edge properness. A valid coloring supplies an upper bound, not automatically equality with the chromatic number. The exact result, retaining the requested order, is ([0, 1, 2, 0, 2, 1], [0, 1, 2], 3, Yes, proper_coloring_only).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r01-chromatic-number-0014/)

Chromatic number · qf-mod37-chromatic-proper-audit · candidate





## 15. Proper Coloring Alone Does Not Prove the Chromatic Minimum

Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 3], [0, 5], [1, 2], [1, 3], [1, 4], [2, 3], [3, 4], [4, 5]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Supplied colors=[0, 1, 0, 2, 0, 1] in vertex order. Color labels are nonnegative integer palette indices; retain their actual labels. Properness checks every edge, and using palette [0,k) is separate from merely using at most k distinct labels. Give (colors, used_colors, color_count, proper, boundary).

View answer and explanation([0, 1, 0, 2, 0, 1], [0, 1, 2], 3, Yes, proper_coloring_only)

Report the original color array, increasingly ordered used labels, their count and all-edge properness. A valid coloring supplies an upper bound, not automatically equality with the chromatic number. The exact result, retaining the requested order, is ([0, 1, 0, 2, 0, 1], [0, 1, 2], 3, Yes, proper_coloring_only).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r01-chromatic-number-0015/)

Chromatic number · qf-mod37-chromatic-proper-audit · candidate





## 16. Proper Coloring Alone Does Not Prove the Chromatic Minimum

Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 3], [0, 5], [1, 2], [1, 3], [1, 4], [2, 3], [3, 4], [4, 5]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Supplied colors=[0, 1, 2, 3, 0, 1] in vertex order. Color labels are nonnegative integer palette indices; retain their actual labels. Properness checks every edge, and using palette [0,k) is separate from merely using at most k distinct labels. Give (colors, used_colors, color_count, proper, boundary).

View answer and explanation([0, 1, 2, 3, 0, 1], [0, 1, 2, 3], 4, Yes, proper_coloring_only)

Report the original color array, increasingly ordered used labels, their count and all-edge properness. A valid coloring supplies an upper bound, not automatically equality with the chromatic number. The exact result, retaining the requested order, is ([0, 1, 2, 3, 0, 1], [0, 1, 2, 3], 4, Yes, proper_coloring_only).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r01-chromatic-number-0016/)

Chromatic number · qf-mod37-chromatic-proper-audit · candidate





## 17. Proper Coloring Alone Does Not Prove the Chromatic Minimum

Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 5], [1, 2], [2, 3], [2, 5], [3, 4], [4, 5]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Supplied colors=[0, 1, 0, 1, 0, 1] in vertex order. Color labels are nonnegative integer palette indices; retain their actual labels. Properness checks every edge, and using palette [0,k) is separate from merely using at most k distinct labels. Give (colors, used_colors, color_count, proper, boundary).

View answer and explanation([0, 1, 0, 1, 0, 1], [0, 1], 2, Yes, proper_coloring_only)

Report the original color array, increasingly ordered used labels, their count and all-edge properness. A valid coloring supplies an upper bound, not automatically equality with the chromatic number. The exact result, retaining the requested order, is ([0, 1, 0, 1, 0, 1], [0, 1], 2, Yes, proper_coloring_only).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r01-chromatic-number-0017/)

Chromatic number · qf-mod37-chromatic-proper-audit · candidate





## 18. Proper Coloring Alone Does Not Prove the Chromatic Minimum

Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 5], [1, 2], [2, 3], [2, 5], [3, 4], [4, 5]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Supplied colors=[0, 1, 2, 0, 2, 1] in vertex order. Color labels are nonnegative integer palette indices; retain their actual labels. Properness checks every edge, and using palette [0,k) is separate from merely using at most k distinct labels. Give (colors, used_colors, color_count, proper, boundary).

View answer and explanation([0, 1, 2, 0, 2, 1], [0, 1, 2], 3, Yes, proper_coloring_only)

Report the original color array, increasingly ordered used labels, their count and all-edge properness. A valid coloring supplies an upper bound, not automatically equality with the chromatic number. The exact result, retaining the requested order, is ([0, 1, 2, 0, 2, 1], [0, 1, 2], 3, Yes, proper_coloring_only).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r01-chromatic-number-0018/)

Chromatic number · qf-mod37-chromatic-proper-audit · candidate





## 19. Proper Coloring Alone Does Not Prove the Chromatic Minimum

Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 3], [0, 5], [1, 2], [2, 3], [2, 5], [3, 4], [4, 5]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Supplied colors=[0, 1, 0, 1, 0, 1] in vertex order. Color labels are nonnegative integer palette indices; retain their actual labels. Properness checks every edge, and using palette [0,k) is separate from merely using at most k distinct labels. Give (colors, used_colors, color_count, proper, boundary).

View answer and explanation([0, 1, 0, 1, 0, 1], [0, 1], 2, Yes, proper_coloring_only)

Report the original color array, increasingly ordered used labels, their count and all-edge properness. A valid coloring supplies an upper bound, not automatically equality with the chromatic number. The exact result, retaining the requested order, is ([0, 1, 0, 1, 0, 1], [0, 1], 2, Yes, proper_coloring_only).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r01-chromatic-number-0019/)

Chromatic number · qf-mod37-chromatic-proper-audit · candidate





## 20. Proper Coloring Alone Does Not Prove the Chromatic Minimum

Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 3], [0, 5], [1, 2], [2, 3], [2, 5], [3, 4], [4, 5]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Supplied colors=[0, 1, 2, 1, 0, 1] in vertex order. Color labels are nonnegative integer palette indices; retain their actual labels. Properness checks every edge, and using palette [0,k) is separate from merely using at most k distinct labels. Give (colors, used_colors, color_count, proper, boundary).

View answer and explanation([0, 1, 2, 1, 0, 1], [0, 1, 2], 3, Yes, proper_coloring_only)

Report the original color array, increasingly ordered used labels, their count and all-edge properness. A valid coloring supplies an upper bound, not automatically equality with the chromatic number. The exact result, retaining the requested order, is ([0, 1, 2, 1, 0, 1], [0, 1, 2], 3, Yes, proper_coloring_only).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r01-chromatic-number-0020/)

Chromatic number · qf-mod37-chromatic-proper-audit · candidate





## 21. Proper Coloring Alone Does Not Prove the Chromatic Minimum

Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 5], [1, 2], [1, 3], [2, 3], [2, 5], [3, 4], [4, 5]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Supplied colors=[0, 1, 0, 2, 0, 1] in vertex order. Color labels are nonnegative integer palette indices; retain their actual labels. Properness checks every edge, and using palette [0,k) is separate from merely using at most k distinct labels. Give (colors, used_colors, color_count, proper, boundary).

View answer and explanation([0, 1, 0, 2, 0, 1], [0, 1, 2], 3, Yes, proper_coloring_only)

Report the original color array, increasingly ordered used labels, their count and all-edge properness. A valid coloring supplies an upper bound, not automatically equality with the chromatic number. The exact result, retaining the requested order, is ([0, 1, 0, 2, 0, 1], [0, 1, 2], 3, Yes, proper_coloring_only).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r01-chromatic-number-0021/)

Chromatic number · qf-mod37-chromatic-proper-audit · candidate





## 22. Proper Coloring Alone Does Not Prove the Chromatic Minimum

Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 5], [1, 2], [1, 3], [2, 3], [2, 5], [3, 4], [4, 5]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Supplied colors=[0, 1, 2, 0, 2, 1] in vertex order. Color labels are nonnegative integer palette indices; retain their actual labels. Properness checks every edge, and using palette [0,k) is separate from merely using at most k distinct labels. Give (colors, used_colors, color_count, proper, boundary).

View answer and explanation([0, 1, 2, 0, 2, 1], [0, 1, 2], 3, Yes, proper_coloring_only)

Report the original color array, increasingly ordered used labels, their count and all-edge properness. A valid coloring supplies an upper bound, not automatically equality with the chromatic number. The exact result, retaining the requested order, is ([0, 1, 2, 0, 2, 1], [0, 1, 2], 3, Yes, proper_coloring_only).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r01-chromatic-number-0022/)

Chromatic number · qf-mod37-chromatic-proper-audit · candidate





## 23. Proper Coloring Alone Does Not Prove the Chromatic Minimum

Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 3], [0, 5], [1, 2], [1, 3], [2, 3], [2, 5], [3, 4], [4, 5]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Supplied colors=[0, 1, 0, 2, 0, 1] in vertex order. Color labels are nonnegative integer palette indices; retain their actual labels. Properness checks every edge, and using palette [0,k) is separate from merely using at most k distinct labels. Give (colors, used_colors, color_count, proper, boundary).

View answer and explanation([0, 1, 0, 2, 0, 1], [0, 1, 2], 3, Yes, proper_coloring_only)

Report the original color array, increasingly ordered used labels, their count and all-edge properness. A valid coloring supplies an upper bound, not automatically equality with the chromatic number. The exact result, retaining the requested order, is ([0, 1, 0, 2, 0, 1], [0, 1, 2], 3, Yes, proper_coloring_only).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r01-chromatic-number-0023/)

Chromatic number · qf-mod37-chromatic-proper-audit · candidate





## 24. Proper Coloring Alone Does Not Prove the Chromatic Minimum

Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 3], [0, 5], [1, 2], [1, 3], [2, 3], [2, 5], [3, 4], [4, 5]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Supplied colors=[0, 1, 2, 3, 0, 1] in vertex order. Color labels are nonnegative integer palette indices; retain their actual labels. Properness checks every edge, and using palette [0,k) is separate from merely using at most k distinct labels. Give (colors, used_colors, color_count, proper, boundary).

View answer and explanation([0, 1, 2, 3, 0, 1], [0, 1, 2, 3], 4, Yes, proper_coloring_only)

Report the original color array, increasingly ordered used labels, their count and all-edge properness. A valid coloring supplies an upper bound, not automatically equality with the chromatic number. The exact result, retaining the requested order, is ([0, 1, 2, 3, 0, 1], [0, 1, 2, 3], 4, Yes, proper_coloring_only).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r01-chromatic-number-0024/)

Chromatic number · qf-mod37-chromatic-proper-audit · candidate





## 25. Proper Coloring Alone Does Not Prove the Chromatic Minimum

Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 5], [1, 2], [1, 4], [2, 3], [2, 5], [3, 4], [4, 5]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Supplied colors=[0, 1, 0, 1, 0, 1] in vertex order. Color labels are nonnegative integer palette indices; retain their actual labels. Properness checks every edge, and using palette [0,k) is separate from merely using at most k distinct labels. Give (colors, used_colors, color_count, proper, boundary).

View answer and explanation([0, 1, 0, 1, 0, 1], [0, 1], 2, Yes, proper_coloring_only)

Report the original color array, increasingly ordered used labels, their count and all-edge properness. A valid coloring supplies an upper bound, not automatically equality with the chromatic number. The exact result, retaining the requested order, is ([0, 1, 0, 1, 0, 1], [0, 1], 2, Yes, proper_coloring_only).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r01-chromatic-number-0025/)

Chromatic number · qf-mod37-chromatic-proper-audit · candidate





## 26. A Supplied k-Palette Is a Candidate Upper Certificate

Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 5], [1, 2], [2, 3], [3, 4], [4, 5]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Supplied colors=[0, 1, 0, 1, 0, 1] in vertex order. Color labels are nonnegative integer palette indices; retain their actual labels. Properness checks every edge, and using palette [0,k) is separate from merely using at most k distinct labels. Requested palette size k=3. Give (k, proper, palette_within_k, candidate_upper_bound_certificate, boundary).

View answer and explanation(3, Yes, Yes, Yes, chi_at_most_k_not_equality)

Check properness and that every label lies below k separately. Their conjunction certifies this supplied palette. A failed candidate does not prove that no other k-coloring exists. The exact result, retaining the requested order, is (3, Yes, Yes, Yes, chi_at_most_k_not_equality).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r01-chromatic-number-0026/)

Chromatic number · qf-mod37-chromatic-upper-bound · candidate





## 27. A Supplied k-Palette Is a Candidate Upper Certificate

Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 5], [1, 2], [2, 3], [3, 4], [4, 5]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Supplied colors=[0, 1, 2, 0, 1, 2] in vertex order. Color labels are nonnegative integer palette indices; retain their actual labels. Properness checks every edge, and using palette [0,k) is separate from merely using at most k distinct labels. Requested palette size k=4. Give (k, proper, palette_within_k, candidate_upper_bound_certificate, boundary).

View answer and explanation(4, Yes, Yes, Yes, chi_at_most_k_not_equality)

Check properness and that every label lies below k separately. Their conjunction certifies this supplied palette. A failed candidate does not prove that no other k-coloring exists. The exact result, retaining the requested order, is (4, Yes, Yes, Yes, chi_at_most_k_not_equality).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r01-chromatic-number-0027/)

Chromatic number · qf-mod37-chromatic-upper-bound · candidate





## 28. A Supplied k-Palette Is a Candidate Upper Certificate

Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 3], [0, 5], [1, 2], [2, 3], [3, 4], [4, 5]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Supplied colors=[0, 1, 0, 1, 0, 1] in vertex order. Color labels are nonnegative integer palette indices; retain their actual labels. Properness checks every edge, and using palette [0,k) is separate from merely using at most k distinct labels. Requested palette size k=3. Give (k, proper, palette_within_k, candidate_upper_bound_certificate, boundary).

View answer and explanation(3, Yes, Yes, Yes, chi_at_most_k_not_equality)

Check properness and that every label lies below k separately. Their conjunction certifies this supplied palette. A failed candidate does not prove that no other k-coloring exists. The exact result, retaining the requested order, is (3, Yes, Yes, Yes, chi_at_most_k_not_equality).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r01-chromatic-number-0028/)

Chromatic number · qf-mod37-chromatic-upper-bound · candidate





## 29. A Supplied k-Palette Is a Candidate Upper Certificate

Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 3], [0, 5], [1, 2], [2, 3], [3, 4], [4, 5]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Supplied colors=[0, 1, 2, 1, 0, 1] in vertex order. Color labels are nonnegative integer palette indices; retain their actual labels. Properness checks every edge, and using palette [0,k) is separate from merely using at most k distinct labels. Requested palette size k=4. Give (k, proper, palette_within_k, candidate_upper_bound_certificate, boundary).

View answer and explanation(4, Yes, Yes, Yes, chi_at_most_k_not_equality)

Check properness and that every label lies below k separately. Their conjunction certifies this supplied palette. A failed candidate does not prove that no other k-coloring exists. The exact result, retaining the requested order, is (4, Yes, Yes, Yes, chi_at_most_k_not_equality).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r01-chromatic-number-0029/)

Chromatic number · qf-mod37-chromatic-upper-bound · candidate





## 30. A Supplied k-Palette Is a Candidate Upper Certificate

Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 5], [1, 2], [1, 3], [2, 3], [3, 4], [4, 5]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Supplied colors=[0, 1, 0, 2, 0, 1] in vertex order. Color labels are nonnegative integer palette indices; retain their actual labels. Properness checks every edge, and using palette [0,k) is separate from merely using at most k distinct labels. Requested palette size k=4. Give (k, proper, palette_within_k, candidate_upper_bound_certificate, boundary).

View answer and explanation(4, Yes, Yes, Yes, chi_at_most_k_not_equality)

Check properness and that every label lies below k separately. Their conjunction certifies this supplied palette. A failed candidate does not prove that no other k-coloring exists. The exact result, retaining the requested order, is (4, Yes, Yes, Yes, chi_at_most_k_not_equality).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r01-chromatic-number-0030/)

Chromatic number · qf-mod37-chromatic-upper-bound · candidate





## 31. A Supplied k-Palette Is a Candidate Upper Certificate

Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 5], [1, 2], [1, 3], [2, 3], [3, 4], [4, 5]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Supplied colors=[0, 1, 2, 0, 1, 2] in vertex order. Color labels are nonnegative integer palette indices; retain their actual labels. Properness checks every edge, and using palette [0,k) is separate from merely using at most k distinct labels. Requested palette size k=4. Give (k, proper, palette_within_k, candidate_upper_bound_certificate, boundary).

View answer and explanation(4, Yes, Yes, Yes, chi_at_most_k_not_equality)

Check properness and that every label lies below k separately. Their conjunction certifies this supplied palette. A failed candidate does not prove that no other k-coloring exists. The exact result, retaining the requested order, is (4, Yes, Yes, Yes, chi_at_most_k_not_equality).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r01-chromatic-number-0031/)

Chromatic number · qf-mod37-chromatic-upper-bound · candidate





## 32. A Supplied k-Palette Is a Candidate Upper Certificate

Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 3], [0, 5], [1, 2], [1, 3], [2, 3], [3, 4], [4, 5]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Supplied colors=[0, 1, 0, 2, 0, 1] in vertex order. Color labels are nonnegative integer palette indices; retain their actual labels. Properness checks every edge, and using palette [0,k) is separate from merely using at most k distinct labels. Requested palette size k=4. Give (k, proper, palette_within_k, candidate_upper_bound_certificate, boundary).

View answer and explanation(4, Yes, Yes, Yes, chi_at_most_k_not_equality)

Check properness and that every label lies below k separately. Their conjunction certifies this supplied palette. A failed candidate does not prove that no other k-coloring exists. The exact result, retaining the requested order, is (4, Yes, Yes, Yes, chi_at_most_k_not_equality).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r01-chromatic-number-0032/)

Chromatic number · qf-mod37-chromatic-upper-bound · candidate





## 33. A Supplied k-Palette Is a Candidate Upper Certificate

Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 3], [0, 5], [1, 2], [1, 3], [2, 3], [3, 4], [4, 5]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Supplied colors=[0, 1, 2, 3, 0, 1] in vertex order. Color labels are nonnegative integer palette indices; retain their actual labels. Properness checks every edge, and using palette [0,k) is separate from merely using at most k distinct labels. Requested palette size k=5. Give (k, proper, palette_within_k, candidate_upper_bound_certificate, boundary).

View answer and explanation(5, Yes, Yes, Yes, chi_at_most_k_not_equality)

Check properness and that every label lies below k separately. Their conjunction certifies this supplied palette. A failed candidate does not prove that no other k-coloring exists. The exact result, retaining the requested order, is (5, Yes, Yes, Yes, chi_at_most_k_not_equality).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r01-chromatic-number-0033/)

Chromatic number · qf-mod37-chromatic-upper-bound · candidate





## 34. A Supplied k-Palette Is a Candidate Upper Certificate

Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 5], [1, 2], [1, 4], [2, 3], [3, 4], [4, 5]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Supplied colors=[0, 1, 0, 1, 0, 1] in vertex order. Color labels are nonnegative integer palette indices; retain their actual labels. Properness checks every edge, and using palette [0,k) is separate from merely using at most k distinct labels. Requested palette size k=3. Give (k, proper, palette_within_k, candidate_upper_bound_certificate, boundary).

View answer and explanation(3, Yes, Yes, Yes, chi_at_most_k_not_equality)

Check properness and that every label lies below k separately. Their conjunction certifies this supplied palette. A failed candidate does not prove that no other k-coloring exists. The exact result, retaining the requested order, is (3, Yes, Yes, Yes, chi_at_most_k_not_equality).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r01-chromatic-number-0034/)

Chromatic number · qf-mod37-chromatic-upper-bound · candidate





## 35. A Supplied k-Palette Is a Candidate Upper Certificate

Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 5], [1, 2], [1, 4], [2, 3], [3, 4], [4, 5]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Supplied colors=[0, 1, 2, 0, 2, 1] in vertex order. Color labels are nonnegative integer palette indices; retain their actual labels. Properness checks every edge, and using palette [0,k) is separate from merely using at most k distinct labels. Requested palette size k=4. Give (k, proper, palette_within_k, candidate_upper_bound_certificate, boundary).

View answer and explanation(4, Yes, Yes, Yes, chi_at_most_k_not_equality)

Check properness and that every label lies below k separately. Their conjunction certifies this supplied palette. A failed candidate does not prove that no other k-coloring exists. The exact result, retaining the requested order, is (4, Yes, Yes, Yes, chi_at_most_k_not_equality).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r01-chromatic-number-0035/)

Chromatic number · qf-mod37-chromatic-upper-bound · candidate





## 36. A Supplied k-Palette Is a Candidate Upper Certificate

Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 3], [0, 5], [1, 2], [1, 4], [2, 3], [3, 4], [4, 5]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Supplied colors=[0, 1, 0, 1, 0, 1] in vertex order. Color labels are nonnegative integer palette indices; retain their actual labels. Properness checks every edge, and using palette [0,k) is separate from merely using at most k distinct labels. Requested palette size k=3. Give (k, proper, palette_within_k, candidate_upper_bound_certificate, boundary).

View answer and explanation(3, Yes, Yes, Yes, chi_at_most_k_not_equality)

Check properness and that every label lies below k separately. Their conjunction certifies this supplied palette. A failed candidate does not prove that no other k-coloring exists. The exact result, retaining the requested order, is (3, Yes, Yes, Yes, chi_at_most_k_not_equality).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r01-chromatic-number-0036/)

Chromatic number · qf-mod37-chromatic-upper-bound · candidate





## 37. A Supplied k-Palette Is a Candidate Upper Certificate

Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 3], [0, 5], [1, 2], [1, 4], [2, 3], [3, 4], [4, 5]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Supplied colors=[0, 1, 2, 1, 0, 1] in vertex order. Color labels are nonnegative integer palette indices; retain their actual labels. Properness checks every edge, and using palette [0,k) is separate from merely using at most k distinct labels. Requested palette size k=4. Give (k, proper, palette_within_k, candidate_upper_bound_certificate, boundary).

View answer and explanation(4, Yes, Yes, Yes, chi_at_most_k_not_equality)

Check properness and that every label lies below k separately. Their conjunction certifies this supplied palette. A failed candidate does not prove that no other k-coloring exists. The exact result, retaining the requested order, is (4, Yes, Yes, Yes, chi_at_most_k_not_equality).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r01-chromatic-number-0037/)

Chromatic number · qf-mod37-chromatic-upper-bound · candidate





## 38. A Supplied k-Palette Is a Candidate Upper Certificate

Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 5], [1, 2], [1, 3], [1, 4], [2, 3], [3, 4], [4, 5]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Supplied colors=[0, 1, 0, 2, 0, 1] in vertex order. Color labels are nonnegative integer palette indices; retain their actual labels. Properness checks every edge, and using palette [0,k) is separate from merely using at most k distinct labels. Requested palette size k=4. Give (k, proper, palette_within_k, candidate_upper_bound_certificate, boundary).

View answer and explanation(4, Yes, Yes, Yes, chi_at_most_k_not_equality)

Check properness and that every label lies below k separately. Their conjunction certifies this supplied palette. A failed candidate does not prove that no other k-coloring exists. The exact result, retaining the requested order, is (4, Yes, Yes, Yes, chi_at_most_k_not_equality).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r01-chromatic-number-0038/)

Chromatic number · qf-mod37-chromatic-upper-bound · candidate





## 39. A Supplied k-Palette Is a Candidate Upper Certificate

Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 5], [1, 2], [1, 3], [1, 4], [2, 3], [3, 4], [4, 5]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Supplied colors=[0, 1, 2, 0, 2, 1] in vertex order. Color labels are nonnegative integer palette indices; retain their actual labels. Properness checks every edge, and using palette [0,k) is separate from merely using at most k distinct labels. Requested palette size k=4. Give (k, proper, palette_within_k, candidate_upper_bound_certificate, boundary).

View answer and explanation(4, Yes, Yes, Yes, chi_at_most_k_not_equality)

Check properness and that every label lies below k separately. Their conjunction certifies this supplied palette. A failed candidate does not prove that no other k-coloring exists. The exact result, retaining the requested order, is (4, Yes, Yes, Yes, chi_at_most_k_not_equality).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r01-chromatic-number-0039/)

Chromatic number · qf-mod37-chromatic-upper-bound · candidate





## 40. A Supplied k-Palette Is a Candidate Upper Certificate

Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 3], [0, 5], [1, 2], [1, 3], [1, 4], [2, 3], [3, 4], [4, 5]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Supplied colors=[0, 1, 0, 2, 0, 1] in vertex order. Color labels are nonnegative integer palette indices; retain their actual labels. Properness checks every edge, and using palette [0,k) is separate from merely using at most k distinct labels. Requested palette size k=4. Give (k, proper, palette_within_k, candidate_upper_bound_certificate, boundary).

View answer and explanation(4, Yes, Yes, Yes, chi_at_most_k_not_equality)

Check properness and that every label lies below k separately. Their conjunction certifies this supplied palette. A failed candidate does not prove that no other k-coloring exists. The exact result, retaining the requested order, is (4, Yes, Yes, Yes, chi_at_most_k_not_equality).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r01-chromatic-number-0040/)

Chromatic number · qf-mod37-chromatic-upper-bound · candidate





## 41. A Supplied k-Palette Is a Candidate Upper Certificate

Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 3], [0, 5], [1, 2], [1, 3], [1, 4], [2, 3], [3, 4], [4, 5]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Supplied colors=[0, 1, 2, 3, 0, 1] in vertex order. Color labels are nonnegative integer palette indices; retain their actual labels. Properness checks every edge, and using palette [0,k) is separate from merely using at most k distinct labels. Requested palette size k=5. Give (k, proper, palette_within_k, candidate_upper_bound_certificate, boundary).

View answer and explanation(5, Yes, Yes, Yes, chi_at_most_k_not_equality)

Check properness and that every label lies below k separately. Their conjunction certifies this supplied palette. A failed candidate does not prove that no other k-coloring exists. The exact result, retaining the requested order, is (5, Yes, Yes, Yes, chi_at_most_k_not_equality).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r01-chromatic-number-0041/)

Chromatic number · qf-mod37-chromatic-upper-bound · candidate





## 42. A Supplied k-Palette Is a Candidate Upper Certificate

Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 5], [1, 2], [2, 3], [2, 5], [3, 4], [4, 5]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Supplied colors=[0, 1, 0, 1, 0, 1] in vertex order. Color labels are nonnegative integer palette indices; retain their actual labels. Properness checks every edge, and using palette [0,k) is separate from merely using at most k distinct labels. Requested palette size k=3. Give (k, proper, palette_within_k, candidate_upper_bound_certificate, boundary).

View answer and explanation(3, Yes, Yes, Yes, chi_at_most_k_not_equality)

Check properness and that every label lies below k separately. Their conjunction certifies this supplied palette. A failed candidate does not prove that no other k-coloring exists. The exact result, retaining the requested order, is (3, Yes, Yes, Yes, chi_at_most_k_not_equality).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r01-chromatic-number-0042/)

Chromatic number · qf-mod37-chromatic-upper-bound · candidate





## 43. A Supplied k-Palette Is a Candidate Upper Certificate

Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 5], [1, 2], [2, 3], [2, 5], [3, 4], [4, 5]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Supplied colors=[0, 1, 2, 0, 2, 1] in vertex order. Color labels are nonnegative integer palette indices; retain their actual labels. Properness checks every edge, and using palette [0,k) is separate from merely using at most k distinct labels. Requested palette size k=4. Give (k, proper, palette_within_k, candidate_upper_bound_certificate, boundary).

View answer and explanation(4, Yes, Yes, Yes, chi_at_most_k_not_equality)

Check properness and that every label lies below k separately. Their conjunction certifies this supplied palette. A failed candidate does not prove that no other k-coloring exists. The exact result, retaining the requested order, is (4, Yes, Yes, Yes, chi_at_most_k_not_equality).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r01-chromatic-number-0043/)

Chromatic number · qf-mod37-chromatic-upper-bound · candidate





## 44. A Supplied k-Palette Is a Candidate Upper Certificate

Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 3], [0, 5], [1, 2], [2, 3], [2, 5], [3, 4], [4, 5]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Supplied colors=[0, 1, 0, 1, 0, 1] in vertex order. Color labels are nonnegative integer palette indices; retain their actual labels. Properness checks every edge, and using palette [0,k) is separate from merely using at most k distinct labels. Requested palette size k=3. Give (k, proper, palette_within_k, candidate_upper_bound_certificate, boundary).

View answer and explanation(3, Yes, Yes, Yes, chi_at_most_k_not_equality)

Check properness and that every label lies below k separately. Their conjunction certifies this supplied palette. A failed candidate does not prove that no other k-coloring exists. The exact result, retaining the requested order, is (3, Yes, Yes, Yes, chi_at_most_k_not_equality).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r01-chromatic-number-0044/)

Chromatic number · qf-mod37-chromatic-upper-bound · candidate





## 45. A Supplied k-Palette Is a Candidate Upper Certificate

Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 3], [0, 5], [1, 2], [2, 3], [2, 5], [3, 4], [4, 5]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Supplied colors=[0, 1, 2, 1, 0, 1] in vertex order. Color labels are nonnegative integer palette indices; retain their actual labels. Properness checks every edge, and using palette [0,k) is separate from merely using at most k distinct labels. Requested palette size k=4. Give (k, proper, palette_within_k, candidate_upper_bound_certificate, boundary).

View answer and explanation(4, Yes, Yes, Yes, chi_at_most_k_not_equality)

Check properness and that every label lies below k separately. Their conjunction certifies this supplied palette. A failed candidate does not prove that no other k-coloring exists. The exact result, retaining the requested order, is (4, Yes, Yes, Yes, chi_at_most_k_not_equality).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r01-chromatic-number-0045/)

Chromatic number · qf-mod37-chromatic-upper-bound · candidate





## 46. A Supplied k-Palette Is a Candidate Upper Certificate

Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 5], [1, 2], [1, 3], [2, 3], [2, 5], [3, 4], [4, 5]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Supplied colors=[0, 1, 0, 2, 0, 1] in vertex order. Color labels are nonnegative integer palette indices; retain their actual labels. Properness checks every edge, and using palette [0,k) is separate from merely using at most k distinct labels. Requested palette size k=4. Give (k, proper, palette_within_k, candidate_upper_bound_certificate, boundary).

View answer and explanation(4, Yes, Yes, Yes, chi_at_most_k_not_equality)

Check properness and that every label lies below k separately. Their conjunction certifies this supplied palette. A failed candidate does not prove that no other k-coloring exists. The exact result, retaining the requested order, is (4, Yes, Yes, Yes, chi_at_most_k_not_equality).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r01-chromatic-number-0046/)

Chromatic number · qf-mod37-chromatic-upper-bound · candidate





## 47. A Supplied k-Palette Is a Candidate Upper Certificate

Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 5], [1, 2], [1, 3], [2, 3], [2, 5], [3, 4], [4, 5]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Supplied colors=[0, 1, 2, 0, 2, 1] in vertex order. Color labels are nonnegative integer palette indices; retain their actual labels. Properness checks every edge, and using palette [0,k) is separate from merely using at most k distinct labels. Requested palette size k=4. Give (k, proper, palette_within_k, candidate_upper_bound_certificate, boundary).

View answer and explanation(4, Yes, Yes, Yes, chi_at_most_k_not_equality)

Check properness and that every label lies below k separately. Their conjunction certifies this supplied palette. A failed candidate does not prove that no other k-coloring exists. The exact result, retaining the requested order, is (4, Yes, Yes, Yes, chi_at_most_k_not_equality).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r01-chromatic-number-0047/)

Chromatic number · qf-mod37-chromatic-upper-bound · candidate





## 48. A Supplied k-Palette Is a Candidate Upper Certificate

Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 3], [0, 5], [1, 2], [1, 3], [2, 3], [2, 5], [3, 4], [4, 5]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Supplied colors=[0, 1, 0, 2, 0, 1] in vertex order. Color labels are nonnegative integer palette indices; retain their actual labels. Properness checks every edge, and using palette [0,k) is separate from merely using at most k distinct labels. Requested palette size k=4. Give (k, proper, palette_within_k, candidate_upper_bound_certificate, boundary).

View answer and explanation(4, Yes, Yes, Yes, chi_at_most_k_not_equality)

Check properness and that every label lies below k separately. Their conjunction certifies this supplied palette. A failed candidate does not prove that no other k-coloring exists. The exact result, retaining the requested order, is (4, Yes, Yes, Yes, chi_at_most_k_not_equality).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r01-chromatic-number-0048/)

Chromatic number · qf-mod37-chromatic-upper-bound · candidate





## 49. A Supplied k-Palette Is a Candidate Upper Certificate

Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 3], [0, 5], [1, 2], [1, 3], [2, 3], [2, 5], [3, 4], [4, 5]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Supplied colors=[0, 1, 2, 3, 0, 1] in vertex order. Color labels are nonnegative integer palette indices; retain their actual labels. Properness checks every edge, and using palette [0,k) is separate from merely using at most k distinct labels. Requested palette size k=5. Give (k, proper, palette_within_k, candidate_upper_bound_certificate, boundary).

View answer and explanation(5, Yes, Yes, Yes, chi_at_most_k_not_equality)

Check properness and that every label lies below k separately. Their conjunction certifies this supplied palette. A failed candidate does not prove that no other k-coloring exists. The exact result, retaining the requested order, is (5, Yes, Yes, Yes, chi_at_most_k_not_equality).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r01-chromatic-number-0049/)

Chromatic number · qf-mod37-chromatic-upper-bound · candidate





## 50. A Supplied k-Palette Is a Candidate Upper Certificate

Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 5], [1, 2], [1, 4], [2, 3], [2, 5], [3, 4], [4, 5]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Supplied colors=[0, 1, 0, 1, 0, 1] in vertex order. Color labels are nonnegative integer palette indices; retain their actual labels. Properness checks every edge, and using palette [0,k) is separate from merely using at most k distinct labels. Requested palette size k=3. Give (k, proper, palette_within_k, candidate_upper_bound_certificate, boundary).

View answer and explanation(3, Yes, Yes, Yes, chi_at_most_k_not_equality)

Check properness and that every label lies below k separately. Their conjunction certifies this supplied palette. A failed candidate does not prove that no other k-coloring exists. The exact result, retaining the requested order, is (3, Yes, Yes, Yes, chi_at_most_k_not_equality).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r01-chromatic-number-0050/)

Chromatic number · qf-mod37-chromatic-upper-bound · candidate





## 51. Exact Chromatic Number Needs Both Lower and Upper Evidence

Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 5], [1, 2], [2, 3], [3, 4], [4, 5]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Give (chromatic_number, canonical_coloring, lower_bound_exhausted, certificate).

View answer and explanation(2, [0, 1, 0, 1, 0, 1], Yes, exhaustive_lower_and_constructive_upper_certificates)

Search increasing k completely, retaining the first lexicographic vertex-order coloring found with colors tried increasingly. A lower palette is exhausted and the displayed proper coloring supplies the upper certificate. The empty carrier has chromatic number0; a nonempty edgeless graph has number1. The exact result, retaining the requested order, is (2, [0, 1, 0, 1, 0, 1], Yes, exhaustive_lower_and_constructive_upper_certificates).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r01-chromatic-number-0051/)

Chromatic number · qf-mod37-chromatic-exact-number · candidate





## 52. Exact Chromatic Number Needs Both Lower and Upper Evidence

Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 5], [1, 2], [2, 3], [3, 4], [4, 5]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Give (chromatic_number, canonical_coloring, lower_bound_exhausted, certificate).

View answer and explanation(3, [0, 1, 2, 0, 1, 2], Yes, exhaustive_lower_and_constructive_upper_certificates)

Search increasing k completely, retaining the first lexicographic vertex-order coloring found with colors tried increasingly. A lower palette is exhausted and the displayed proper coloring supplies the upper certificate. The empty carrier has chromatic number0; a nonempty edgeless graph has number1. The exact result, retaining the requested order, is (3, [0, 1, 2, 0, 1, 2], Yes, exhaustive_lower_and_constructive_upper_certificates).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r01-chromatic-number-0052/)

Chromatic number · qf-mod37-chromatic-exact-number · candidate





## 53. Exact Chromatic Number Needs Both Lower and Upper Evidence

Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 3], [0, 5], [1, 2], [2, 3], [3, 4], [4, 5]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Give (chromatic_number, canonical_coloring, lower_bound_exhausted, certificate).

View answer and explanation(2, [0, 1, 0, 1, 0, 1], Yes, exhaustive_lower_and_constructive_upper_certificates)

Search increasing k completely, retaining the first lexicographic vertex-order coloring found with colors tried increasingly. A lower palette is exhausted and the displayed proper coloring supplies the upper certificate. The empty carrier has chromatic number0; a nonempty edgeless graph has number1. The exact result, retaining the requested order, is (2, [0, 1, 0, 1, 0, 1], Yes, exhaustive_lower_and_constructive_upper_certificates).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r01-chromatic-number-0053/)

Chromatic number · qf-mod37-chromatic-exact-number · candidate





## 54. Exact Chromatic Number Needs Both Lower and Upper Evidence

Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 3], [0, 5], [1, 2], [2, 3], [3, 4], [4, 5]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Give (chromatic_number, canonical_coloring, lower_bound_exhausted, certificate).

View answer and explanation(3, [0, 1, 2, 1, 0, 1], Yes, exhaustive_lower_and_constructive_upper_certificates)

Search increasing k completely, retaining the first lexicographic vertex-order coloring found with colors tried increasingly. A lower palette is exhausted and the displayed proper coloring supplies the upper certificate. The empty carrier has chromatic number0; a nonempty edgeless graph has number1. The exact result, retaining the requested order, is (3, [0, 1, 2, 1, 0, 1], Yes, exhaustive_lower_and_constructive_upper_certificates).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r01-chromatic-number-0054/)

Chromatic number · qf-mod37-chromatic-exact-number · candidate





## 55. Exact Chromatic Number Needs Both Lower and Upper Evidence

Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 5], [1, 2], [1, 3], [2, 3], [3, 4], [4, 5]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Give (chromatic_number, canonical_coloring, lower_bound_exhausted, certificate).

View answer and explanation(3, [0, 1, 0, 2, 0, 1], Yes, exhaustive_lower_and_constructive_upper_certificates)

Search increasing k completely, retaining the first lexicographic vertex-order coloring found with colors tried increasingly. A lower palette is exhausted and the displayed proper coloring supplies the upper certificate. The empty carrier has chromatic number0; a nonempty edgeless graph has number1. The exact result, retaining the requested order, is (3, [0, 1, 0, 2, 0, 1], Yes, exhaustive_lower_and_constructive_upper_certificates).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r01-chromatic-number-0055/)

Chromatic number · qf-mod37-chromatic-exact-number · candidate





## 56. Exact Chromatic Number Needs Both Lower and Upper Evidence

Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 5], [1, 2], [1, 3], [2, 3], [3, 4], [4, 5]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Give (chromatic_number, canonical_coloring, lower_bound_exhausted, certificate).

View answer and explanation(3, [0, 1, 2, 0, 1, 2], Yes, exhaustive_lower_and_constructive_upper_certificates)

Search increasing k completely, retaining the first lexicographic vertex-order coloring found with colors tried increasingly. A lower palette is exhausted and the displayed proper coloring supplies the upper certificate. The empty carrier has chromatic number0; a nonempty edgeless graph has number1. The exact result, retaining the requested order, is (3, [0, 1, 2, 0, 1, 2], Yes, exhaustive_lower_and_constructive_upper_certificates).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r01-chromatic-number-0056/)

Chromatic number · qf-mod37-chromatic-exact-number · candidate





## 57. Exact Chromatic Number Needs Both Lower and Upper Evidence

Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 3], [0, 5], [1, 2], [1, 3], [2, 3], [3, 4], [4, 5]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Give (chromatic_number, canonical_coloring, lower_bound_exhausted, certificate).

View answer and explanation(3, [0, 1, 0, 2, 0, 1], Yes, exhaustive_lower_and_constructive_upper_certificates)

Search increasing k completely, retaining the first lexicographic vertex-order coloring found with colors tried increasingly. A lower palette is exhausted and the displayed proper coloring supplies the upper certificate. The empty carrier has chromatic number0; a nonempty edgeless graph has number1. The exact result, retaining the requested order, is (3, [0, 1, 0, 2, 0, 1], Yes, exhaustive_lower_and_constructive_upper_certificates).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r01-chromatic-number-0057/)

Chromatic number · qf-mod37-chromatic-exact-number · candidate





## 58. Exact Chromatic Number Needs Both Lower and Upper Evidence

Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 3], [0, 5], [1, 2], [1, 3], [2, 3], [3, 4], [4, 5]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Give (chromatic_number, canonical_coloring, lower_bound_exhausted, certificate).

View answer and explanation(4, [0, 1, 2, 3, 0, 1], Yes, exhaustive_lower_and_constructive_upper_certificates)

Search increasing k completely, retaining the first lexicographic vertex-order coloring found with colors tried increasingly. A lower palette is exhausted and the displayed proper coloring supplies the upper certificate. The empty carrier has chromatic number0; a nonempty edgeless graph has number1. The exact result, retaining the requested order, is (4, [0, 1, 2, 3, 0, 1], Yes, exhaustive_lower_and_constructive_upper_certificates).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r01-chromatic-number-0058/)

Chromatic number · qf-mod37-chromatic-exact-number · candidate





## 59. Exact Chromatic Number Needs Both Lower and Upper Evidence

Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 5], [1, 2], [1, 4], [2, 3], [3, 4], [4, 5]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Give (chromatic_number, canonical_coloring, lower_bound_exhausted, certificate).

View answer and explanation(2, [0, 1, 0, 1, 0, 1], Yes, exhaustive_lower_and_constructive_upper_certificates)

Search increasing k completely, retaining the first lexicographic vertex-order coloring found with colors tried increasingly. A lower palette is exhausted and the displayed proper coloring supplies the upper certificate. The empty carrier has chromatic number0; a nonempty edgeless graph has number1. The exact result, retaining the requested order, is (2, [0, 1, 0, 1, 0, 1], Yes, exhaustive_lower_and_constructive_upper_certificates).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r01-chromatic-number-0059/)

Chromatic number · qf-mod37-chromatic-exact-number · candidate





## 60. Exact Chromatic Number Needs Both Lower and Upper Evidence

Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 5], [1, 2], [1, 4], [2, 3], [3, 4], [4, 5]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Give (chromatic_number, canonical_coloring, lower_bound_exhausted, certificate).

View answer and explanation(3, [0, 1, 2, 0, 2, 1], Yes, exhaustive_lower_and_constructive_upper_certificates)

Search increasing k completely, retaining the first lexicographic vertex-order coloring found with colors tried increasingly. A lower palette is exhausted and the displayed proper coloring supplies the upper certificate. The empty carrier has chromatic number0; a nonempty edgeless graph has number1. The exact result, retaining the requested order, is (3, [0, 1, 2, 0, 2, 1], Yes, exhaustive_lower_and_constructive_upper_certificates).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r01-chromatic-number-0060/)

Chromatic number · qf-mod37-chromatic-exact-number · candidate





## 61. Exact Chromatic Number Needs Both Lower and Upper Evidence

Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 3], [0, 5], [1, 2], [1, 4], [2, 3], [3, 4], [4, 5]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Give (chromatic_number, canonical_coloring, lower_bound_exhausted, certificate).

View answer and explanation(2, [0, 1, 0, 1, 0, 1], Yes, exhaustive_lower_and_constructive_upper_certificates)

Search increasing k completely, retaining the first lexicographic vertex-order coloring found with colors tried increasingly. A lower palette is exhausted and the displayed proper coloring supplies the upper certificate. The empty carrier has chromatic number0; a nonempty edgeless graph has number1. The exact result, retaining the requested order, is (2, [0, 1, 0, 1, 0, 1], Yes, exhaustive_lower_and_constructive_upper_certificates).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r01-chromatic-number-0061/)

Chromatic number · qf-mod37-chromatic-exact-number · candidate





## 62. Exact Chromatic Number Needs Both Lower and Upper Evidence

Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 3], [0, 5], [1, 2], [1, 4], [2, 3], [3, 4], [4, 5]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Give (chromatic_number, canonical_coloring, lower_bound_exhausted, certificate).

View answer and explanation(3, [0, 1, 2, 1, 0, 1], Yes, exhaustive_lower_and_constructive_upper_certificates)

Search increasing k completely, retaining the first lexicographic vertex-order coloring found with colors tried increasingly. A lower palette is exhausted and the displayed proper coloring supplies the upper certificate. The empty carrier has chromatic number0; a nonempty edgeless graph has number1. The exact result, retaining the requested order, is (3, [0, 1, 2, 1, 0, 1], Yes, exhaustive_lower_and_constructive_upper_certificates).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r01-chromatic-number-0062/)

Chromatic number · qf-mod37-chromatic-exact-number · candidate





## 63. Exact Chromatic Number Needs Both Lower and Upper Evidence

Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 5], [1, 2], [1, 3], [1, 4], [2, 3], [3, 4], [4, 5]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Give (chromatic_number, canonical_coloring, lower_bound_exhausted, certificate).

View answer and explanation(3, [0, 1, 0, 2, 0, 1], Yes, exhaustive_lower_and_constructive_upper_certificates)

Search increasing k completely, retaining the first lexicographic vertex-order coloring found with colors tried increasingly. A lower palette is exhausted and the displayed proper coloring supplies the upper certificate. The empty carrier has chromatic number0; a nonempty edgeless graph has number1. The exact result, retaining the requested order, is (3, [0, 1, 0, 2, 0, 1], Yes, exhaustive_lower_and_constructive_upper_certificates).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r01-chromatic-number-0063/)

Chromatic number · qf-mod37-chromatic-exact-number · candidate





## 64. Exact Chromatic Number Needs Both Lower and Upper Evidence

Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 5], [1, 2], [1, 3], [1, 4], [2, 3], [3, 4], [4, 5]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Give (chromatic_number, canonical_coloring, lower_bound_exhausted, certificate).

View answer and explanation(3, [0, 1, 2, 0, 2, 1], Yes, exhaustive_lower_and_constructive_upper_certificates)

Search increasing k completely, retaining the first lexicographic vertex-order coloring found with colors tried increasingly. A lower palette is exhausted and the displayed proper coloring supplies the upper certificate. The empty carrier has chromatic number0; a nonempty edgeless graph has number1. The exact result, retaining the requested order, is (3, [0, 1, 2, 0, 2, 1], Yes, exhaustive_lower_and_constructive_upper_certificates).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r01-chromatic-number-0064/)

Chromatic number · qf-mod37-chromatic-exact-number · candidate





## 65. Exact Chromatic Number Needs Both Lower and Upper Evidence

Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 3], [0, 5], [1, 2], [1, 3], [1, 4], [2, 3], [3, 4], [4, 5]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Give (chromatic_number, canonical_coloring, lower_bound_exhausted, certificate).

View answer and explanation(3, [0, 1, 0, 2, 0, 1], Yes, exhaustive_lower_and_constructive_upper_certificates)

Search increasing k completely, retaining the first lexicographic vertex-order coloring found with colors tried increasingly. A lower palette is exhausted and the displayed proper coloring supplies the upper certificate. The empty carrier has chromatic number0; a nonempty edgeless graph has number1. The exact result, retaining the requested order, is (3, [0, 1, 0, 2, 0, 1], Yes, exhaustive_lower_and_constructive_upper_certificates).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r01-chromatic-number-0065/)

Chromatic number · qf-mod37-chromatic-exact-number · candidate





## 66. Exact Chromatic Number Needs Both Lower and Upper Evidence

Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 3], [0, 5], [1, 2], [1, 3], [1, 4], [2, 3], [3, 4], [4, 5]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Give (chromatic_number, canonical_coloring, lower_bound_exhausted, certificate).

View answer and explanation(4, [0, 1, 2, 3, 0, 1], Yes, exhaustive_lower_and_constructive_upper_certificates)

Search increasing k completely, retaining the first lexicographic vertex-order coloring found with colors tried increasingly. A lower palette is exhausted and the displayed proper coloring supplies the upper certificate. The empty carrier has chromatic number0; a nonempty edgeless graph has number1. The exact result, retaining the requested order, is (4, [0, 1, 2, 3, 0, 1], Yes, exhaustive_lower_and_constructive_upper_certificates).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r01-chromatic-number-0066/)

Chromatic number · qf-mod37-chromatic-exact-number · candidate





## 67. Exact Chromatic Number Needs Both Lower and Upper Evidence

Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 5], [1, 2], [2, 3], [2, 5], [3, 4], [4, 5]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Give (chromatic_number, canonical_coloring, lower_bound_exhausted, certificate).

View answer and explanation(2, [0, 1, 0, 1, 0, 1], Yes, exhaustive_lower_and_constructive_upper_certificates)

Search increasing k completely, retaining the first lexicographic vertex-order coloring found with colors tried increasingly. A lower palette is exhausted and the displayed proper coloring supplies the upper certificate. The empty carrier has chromatic number0; a nonempty edgeless graph has number1. The exact result, retaining the requested order, is (2, [0, 1, 0, 1, 0, 1], Yes, exhaustive_lower_and_constructive_upper_certificates).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r01-chromatic-number-0067/)

Chromatic number · qf-mod37-chromatic-exact-number · candidate





## 68. Exact Chromatic Number Needs Both Lower and Upper Evidence

Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 5], [1, 2], [2, 3], [2, 5], [3, 4], [4, 5]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Give (chromatic_number, canonical_coloring, lower_bound_exhausted, certificate).

View answer and explanation(3, [0, 1, 2, 0, 2, 1], Yes, exhaustive_lower_and_constructive_upper_certificates)

Search increasing k completely, retaining the first lexicographic vertex-order coloring found with colors tried increasingly. A lower palette is exhausted and the displayed proper coloring supplies the upper certificate. The empty carrier has chromatic number0; a nonempty edgeless graph has number1. The exact result, retaining the requested order, is (3, [0, 1, 2, 0, 2, 1], Yes, exhaustive_lower_and_constructive_upper_certificates).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r01-chromatic-number-0068/)

Chromatic number · qf-mod37-chromatic-exact-number · candidate





## 69. Exact Chromatic Number Needs Both Lower and Upper Evidence

Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 3], [0, 5], [1, 2], [2, 3], [2, 5], [3, 4], [4, 5]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Give (chromatic_number, canonical_coloring, lower_bound_exhausted, certificate).

View answer and explanation(2, [0, 1, 0, 1, 0, 1], Yes, exhaustive_lower_and_constructive_upper_certificates)

Search increasing k completely, retaining the first lexicographic vertex-order coloring found with colors tried increasingly. A lower palette is exhausted and the displayed proper coloring supplies the upper certificate. The empty carrier has chromatic number0; a nonempty edgeless graph has number1. The exact result, retaining the requested order, is (2, [0, 1, 0, 1, 0, 1], Yes, exhaustive_lower_and_constructive_upper_certificates).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r01-chromatic-number-0069/)

Chromatic number · qf-mod37-chromatic-exact-number · candidate





## 70. Exact Chromatic Number Needs Both Lower and Upper Evidence

Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 3], [0, 5], [1, 2], [2, 3], [2, 5], [3, 4], [4, 5]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Give (chromatic_number, canonical_coloring, lower_bound_exhausted, certificate).

View answer and explanation(3, [0, 1, 2, 1, 0, 1], Yes, exhaustive_lower_and_constructive_upper_certificates)

Search increasing k completely, retaining the first lexicographic vertex-order coloring found with colors tried increasingly. A lower palette is exhausted and the displayed proper coloring supplies the upper certificate. The empty carrier has chromatic number0; a nonempty edgeless graph has number1. The exact result, retaining the requested order, is (3, [0, 1, 2, 1, 0, 1], Yes, exhaustive_lower_and_constructive_upper_certificates).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r01-chromatic-number-0070/)

Chromatic number · qf-mod37-chromatic-exact-number · candidate





## 71. Exact Chromatic Number Needs Both Lower and Upper Evidence

Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 5], [1, 2], [1, 3], [2, 3], [2, 5], [3, 4], [4, 5]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Give (chromatic_number, canonical_coloring, lower_bound_exhausted, certificate).

View answer and explanation(3, [0, 1, 0, 2, 0, 1], Yes, exhaustive_lower_and_constructive_upper_certificates)

Search increasing k completely, retaining the first lexicographic vertex-order coloring found with colors tried increasingly. A lower palette is exhausted and the displayed proper coloring supplies the upper certificate. The empty carrier has chromatic number0; a nonempty edgeless graph has number1. The exact result, retaining the requested order, is (3, [0, 1, 0, 2, 0, 1], Yes, exhaustive_lower_and_constructive_upper_certificates).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r01-chromatic-number-0071/)

Chromatic number · qf-mod37-chromatic-exact-number · candidate





## 72. Exact Chromatic Number Needs Both Lower and Upper Evidence

Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 5], [1, 2], [1, 3], [2, 3], [2, 5], [3, 4], [4, 5]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Give (chromatic_number, canonical_coloring, lower_bound_exhausted, certificate).

View answer and explanation(3, [0, 1, 2, 0, 2, 1], Yes, exhaustive_lower_and_constructive_upper_certificates)

Search increasing k completely, retaining the first lexicographic vertex-order coloring found with colors tried increasingly. A lower palette is exhausted and the displayed proper coloring supplies the upper certificate. The empty carrier has chromatic number0; a nonempty edgeless graph has number1. The exact result, retaining the requested order, is (3, [0, 1, 2, 0, 2, 1], Yes, exhaustive_lower_and_constructive_upper_certificates).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r01-chromatic-number-0072/)

Chromatic number · qf-mod37-chromatic-exact-number · candidate





## 73. Exact Chromatic Number Needs Both Lower and Upper Evidence

Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 3], [0, 5], [1, 2], [1, 3], [2, 3], [2, 5], [3, 4], [4, 5]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Give (chromatic_number, canonical_coloring, lower_bound_exhausted, certificate).

View answer and explanation(3, [0, 1, 0, 2, 0, 1], Yes, exhaustive_lower_and_constructive_upper_certificates)

Search increasing k completely, retaining the first lexicographic vertex-order coloring found with colors tried increasingly. A lower palette is exhausted and the displayed proper coloring supplies the upper certificate. The empty carrier has chromatic number0; a nonempty edgeless graph has number1. The exact result, retaining the requested order, is (3, [0, 1, 0, 2, 0, 1], Yes, exhaustive_lower_and_constructive_upper_certificates).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r01-chromatic-number-0073/)

Chromatic number · qf-mod37-chromatic-exact-number · candidate





## 74. Exact Chromatic Number Needs Both Lower and Upper Evidence

Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 3], [0, 5], [1, 2], [1, 3], [2, 3], [2, 5], [3, 4], [4, 5]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Give (chromatic_number, canonical_coloring, lower_bound_exhausted, certificate).

View answer and explanation(4, [0, 1, 2, 3, 0, 1], Yes, exhaustive_lower_and_constructive_upper_certificates)

Search increasing k completely, retaining the first lexicographic vertex-order coloring found with colors tried increasingly. A lower palette is exhausted and the displayed proper coloring supplies the upper certificate. The empty carrier has chromatic number0; a nonempty edgeless graph has number1. The exact result, retaining the requested order, is (4, [0, 1, 2, 3, 0, 1], Yes, exhaustive_lower_and_constructive_upper_certificates).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r01-chromatic-number-0074/)

Chromatic number · qf-mod37-chromatic-exact-number · candidate





## 75. Exact Chromatic Number Needs Both Lower and Upper Evidence

Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 5], [1, 2], [1, 4], [2, 3], [2, 5], [3, 4], [4, 5]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Give (chromatic_number, canonical_coloring, lower_bound_exhausted, certificate).

View answer and explanation(2, [0, 1, 0, 1, 0, 1], Yes, exhaustive_lower_and_constructive_upper_certificates)

Search increasing k completely, retaining the first lexicographic vertex-order coloring found with colors tried increasingly. A lower palette is exhausted and the displayed proper coloring supplies the upper certificate. The empty carrier has chromatic number0; a nonempty edgeless graph has number1. The exact result, retaining the requested order, is (2, [0, 1, 0, 1, 0, 1], Yes, exhaustive_lower_and_constructive_upper_certificates).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r01-chromatic-number-0075/)

Chromatic number · qf-mod37-chromatic-exact-number · candidate





## 76. Equality of k with the Minimum Is Not Validity of This Coloring

Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 5], [1, 2], [2, 3], [3, 4], [4, 5]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Supplied colors=[0, 1, 0, 1, 0, 1] in vertex order. Color labels are nonnegative integer palette indices; retain their actual labels. Properness checks every edge, and using palette [0,k) is separate from merely using at most k distinct labels. Compare k=3. Give (k, candidate_k_palette_valid, chromatic_number, k_equals_chromatic_number, boundary).

k_equals_chromatic_number compares k with the true chromatic number; it does not certify the supplied coloring. Check candidate_k_palette_valid separately.

View answer and explanation(3, Yes, 2, No, k_colorable_means_at_most_k_and_greedy_not_optimality_proof)

Keep the supplied palette validity flag separate from the exhaustive chromatic number and the comparison k=chi. The comparison may be Yes even when the supplied coloring fails. A greedy or arbitrary valid palette alone is not a lower-bound certificate. The exact result, retaining the requested order, is (3, Yes, 2, No, k_colorable_means_at_most_k_and_greedy_not_optimality_proof).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r01-chromatic-number-0076/)

Chromatic number · qf-mod37-chromatic-minimum-boundary · candidate





## 77. Equality of k with the Minimum Is Not Validity of This Coloring

Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 5], [1, 2], [2, 3], [3, 4], [4, 5]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Supplied colors=[0, 1, 2, 0, 1, 2] in vertex order. Color labels are nonnegative integer palette indices; retain their actual labels. Properness checks every edge, and using palette [0,k) is separate from merely using at most k distinct labels. Compare k=4. Give (k, candidate_k_palette_valid, chromatic_number, k_equals_chromatic_number, boundary).

k_equals_chromatic_number compares k with the true chromatic number; it does not certify the supplied coloring. Check candidate_k_palette_valid separately.

View answer and explanation(4, Yes, 3, No, k_colorable_means_at_most_k_and_greedy_not_optimality_proof)

Keep the supplied palette validity flag separate from the exhaustive chromatic number and the comparison k=chi. The comparison may be Yes even when the supplied coloring fails. A greedy or arbitrary valid palette alone is not a lower-bound certificate. The exact result, retaining the requested order, is (4, Yes, 3, No, k_colorable_means_at_most_k_and_greedy_not_optimality_proof).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r01-chromatic-number-0077/)

Chromatic number · qf-mod37-chromatic-minimum-boundary · candidate





## 78. Equality of k with the Minimum Is Not Validity of This Coloring

Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 3], [0, 5], [1, 2], [2, 3], [3, 4], [4, 5]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Supplied colors=[0, 1, 0, 1, 0, 1] in vertex order. Color labels are nonnegative integer palette indices; retain their actual labels. Properness checks every edge, and using palette [0,k) is separate from merely using at most k distinct labels. Compare k=3. Give (k, candidate_k_palette_valid, chromatic_number, k_equals_chromatic_number, boundary).

k_equals_chromatic_number compares k with the true chromatic number; it does not certify the supplied coloring. Check candidate_k_palette_valid separately.

View answer and explanation(3, Yes, 2, No, k_colorable_means_at_most_k_and_greedy_not_optimality_proof)

Keep the supplied palette validity flag separate from the exhaustive chromatic number and the comparison k=chi. The comparison may be Yes even when the supplied coloring fails. A greedy or arbitrary valid palette alone is not a lower-bound certificate. The exact result, retaining the requested order, is (3, Yes, 2, No, k_colorable_means_at_most_k_and_greedy_not_optimality_proof).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r01-chromatic-number-0078/)

Chromatic number · qf-mod37-chromatic-minimum-boundary · candidate





## 79. Equality of k with the Minimum Is Not Validity of This Coloring

Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 3], [0, 5], [1, 2], [2, 3], [3, 4], [4, 5]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Supplied colors=[0, 1, 2, 1, 0, 1] in vertex order. Color labels are nonnegative integer palette indices; retain their actual labels. Properness checks every edge, and using palette [0,k) is separate from merely using at most k distinct labels. Compare k=4. Give (k, candidate_k_palette_valid, chromatic_number, k_equals_chromatic_number, boundary).

k_equals_chromatic_number compares k with the true chromatic number; it does not certify the supplied coloring. Check candidate_k_palette_valid separately.

View answer and explanation(4, Yes, 3, No, k_colorable_means_at_most_k_and_greedy_not_optimality_proof)

Keep the supplied palette validity flag separate from the exhaustive chromatic number and the comparison k=chi. The comparison may be Yes even when the supplied coloring fails. A greedy or arbitrary valid palette alone is not a lower-bound certificate. The exact result, retaining the requested order, is (4, Yes, 3, No, k_colorable_means_at_most_k_and_greedy_not_optimality_proof).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r01-chromatic-number-0079/)

Chromatic number · qf-mod37-chromatic-minimum-boundary · candidate





## 80. Equality of k with the Minimum Is Not Validity of This Coloring

Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 5], [1, 2], [1, 3], [2, 3], [3, 4], [4, 5]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Supplied colors=[0, 1, 0, 2, 0, 1] in vertex order. Color labels are nonnegative integer palette indices; retain their actual labels. Properness checks every edge, and using palette [0,k) is separate from merely using at most k distinct labels. Compare k=4. Give (k, candidate_k_palette_valid, chromatic_number, k_equals_chromatic_number, boundary).

k_equals_chromatic_number compares k with the true chromatic number; it does not certify the supplied coloring. Check candidate_k_palette_valid separately.

View answer and explanation(4, Yes, 3, No, k_colorable_means_at_most_k_and_greedy_not_optimality_proof)

Keep the supplied palette validity flag separate from the exhaustive chromatic number and the comparison k=chi. The comparison may be Yes even when the supplied coloring fails. A greedy or arbitrary valid palette alone is not a lower-bound certificate. The exact result, retaining the requested order, is (4, Yes, 3, No, k_colorable_means_at_most_k_and_greedy_not_optimality_proof).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r01-chromatic-number-0080/)

Chromatic number · qf-mod37-chromatic-minimum-boundary · candidate





## 81. Equality of k with the Minimum Is Not Validity of This Coloring

Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 5], [1, 2], [1, 3], [2, 3], [3, 4], [4, 5]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Supplied colors=[0, 1, 2, 0, 1, 2] in vertex order. Color labels are nonnegative integer palette indices; retain their actual labels. Properness checks every edge, and using palette [0,k) is separate from merely using at most k distinct labels. Compare k=4. Give (k, candidate_k_palette_valid, chromatic_number, k_equals_chromatic_number, boundary).

k_equals_chromatic_number compares k with the true chromatic number; it does not certify the supplied coloring. Check candidate_k_palette_valid separately.

View answer and explanation(4, Yes, 3, No, k_colorable_means_at_most_k_and_greedy_not_optimality_proof)

Keep the supplied palette validity flag separate from the exhaustive chromatic number and the comparison k=chi. The comparison may be Yes even when the supplied coloring fails. A greedy or arbitrary valid palette alone is not a lower-bound certificate. The exact result, retaining the requested order, is (4, Yes, 3, No, k_colorable_means_at_most_k_and_greedy_not_optimality_proof).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r01-chromatic-number-0081/)

Chromatic number · qf-mod37-chromatic-minimum-boundary · candidate





## 82. Equality of k with the Minimum Is Not Validity of This Coloring

Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 3], [0, 5], [1, 2], [1, 3], [2, 3], [3, 4], [4, 5]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Supplied colors=[0, 1, 0, 2, 0, 1] in vertex order. Color labels are nonnegative integer palette indices; retain their actual labels. Properness checks every edge, and using palette [0,k) is separate from merely using at most k distinct labels. Compare k=4. Give (k, candidate_k_palette_valid, chromatic_number, k_equals_chromatic_number, boundary).

k_equals_chromatic_number compares k with the true chromatic number; it does not certify the supplied coloring. Check candidate_k_palette_valid separately.

View answer and explanation(4, Yes, 3, No, k_colorable_means_at_most_k_and_greedy_not_optimality_proof)

Keep the supplied palette validity flag separate from the exhaustive chromatic number and the comparison k=chi. The comparison may be Yes even when the supplied coloring fails. A greedy or arbitrary valid palette alone is not a lower-bound certificate. The exact result, retaining the requested order, is (4, Yes, 3, No, k_colorable_means_at_most_k_and_greedy_not_optimality_proof).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r01-chromatic-number-0082/)

Chromatic number · qf-mod37-chromatic-minimum-boundary · candidate





## 83. Equality of k with the Minimum Is Not Validity of This Coloring

Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 3], [0, 5], [1, 2], [1, 3], [2, 3], [3, 4], [4, 5]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Supplied colors=[0, 1, 2, 3, 0, 1] in vertex order. Color labels are nonnegative integer palette indices; retain their actual labels. Properness checks every edge, and using palette [0,k) is separate from merely using at most k distinct labels. Compare k=5. Give (k, candidate_k_palette_valid, chromatic_number, k_equals_chromatic_number, boundary).

k_equals_chromatic_number compares k with the true chromatic number; it does not certify the supplied coloring. Check candidate_k_palette_valid separately.

View answer and explanation(5, Yes, 4, No, k_colorable_means_at_most_k_and_greedy_not_optimality_proof)

Keep the supplied palette validity flag separate from the exhaustive chromatic number and the comparison k=chi. The comparison may be Yes even when the supplied coloring fails. A greedy or arbitrary valid palette alone is not a lower-bound certificate. The exact result, retaining the requested order, is (5, Yes, 4, No, k_colorable_means_at_most_k_and_greedy_not_optimality_proof).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r01-chromatic-number-0083/)

Chromatic number · qf-mod37-chromatic-minimum-boundary · candidate





## 84. Equality of k with the Minimum Is Not Validity of This Coloring

Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 5], [1, 2], [1, 4], [2, 3], [3, 4], [4, 5]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Supplied colors=[0, 1, 0, 1, 0, 1] in vertex order. Color labels are nonnegative integer palette indices; retain their actual labels. Properness checks every edge, and using palette [0,k) is separate from merely using at most k distinct labels. Compare k=3. Give (k, candidate_k_palette_valid, chromatic_number, k_equals_chromatic_number, boundary).

k_equals_chromatic_number compares k with the true chromatic number; it does not certify the supplied coloring. Check candidate_k_palette_valid separately.

View answer and explanation(3, Yes, 2, No, k_colorable_means_at_most_k_and_greedy_not_optimality_proof)

Keep the supplied palette validity flag separate from the exhaustive chromatic number and the comparison k=chi. The comparison may be Yes even when the supplied coloring fails. A greedy or arbitrary valid palette alone is not a lower-bound certificate. The exact result, retaining the requested order, is (3, Yes, 2, No, k_colorable_means_at_most_k_and_greedy_not_optimality_proof).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r01-chromatic-number-0084/)

Chromatic number · qf-mod37-chromatic-minimum-boundary · candidate





## 85. Equality of k with the Minimum Is Not Validity of This Coloring

Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 5], [1, 2], [1, 4], [2, 3], [3, 4], [4, 5]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Supplied colors=[0, 1, 2, 0, 2, 1] in vertex order. Color labels are nonnegative integer palette indices; retain their actual labels. Properness checks every edge, and using palette [0,k) is separate from merely using at most k distinct labels. Compare k=4. Give (k, candidate_k_palette_valid, chromatic_number, k_equals_chromatic_number, boundary).

k_equals_chromatic_number compares k with the true chromatic number; it does not certify the supplied coloring. Check candidate_k_palette_valid separately.

View answer and explanation(4, Yes, 3, No, k_colorable_means_at_most_k_and_greedy_not_optimality_proof)

Keep the supplied palette validity flag separate from the exhaustive chromatic number and the comparison k=chi. The comparison may be Yes even when the supplied coloring fails. A greedy or arbitrary valid palette alone is not a lower-bound certificate. The exact result, retaining the requested order, is (4, Yes, 3, No, k_colorable_means_at_most_k_and_greedy_not_optimality_proof).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r01-chromatic-number-0085/)

Chromatic number · qf-mod37-chromatic-minimum-boundary · candidate





## 86. Equality of k with the Minimum Is Not Validity of This Coloring

Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 3], [0, 5], [1, 2], [1, 4], [2, 3], [3, 4], [4, 5]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Supplied colors=[0, 1, 0, 1, 0, 1] in vertex order. Color labels are nonnegative integer palette indices; retain their actual labels. Properness checks every edge, and using palette [0,k) is separate from merely using at most k distinct labels. Compare k=3. Give (k, candidate_k_palette_valid, chromatic_number, k_equals_chromatic_number, boundary).

k_equals_chromatic_number compares k with the true chromatic number; it does not certify the supplied coloring. Check candidate_k_palette_valid separately.

View answer and explanation(3, Yes, 2, No, k_colorable_means_at_most_k_and_greedy_not_optimality_proof)

Keep the supplied palette validity flag separate from the exhaustive chromatic number and the comparison k=chi. The comparison may be Yes even when the supplied coloring fails. A greedy or arbitrary valid palette alone is not a lower-bound certificate. The exact result, retaining the requested order, is (3, Yes, 2, No, k_colorable_means_at_most_k_and_greedy_not_optimality_proof).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r01-chromatic-number-0086/)

Chromatic number · qf-mod37-chromatic-minimum-boundary · candidate





## 87. Equality of k with the Minimum Is Not Validity of This Coloring

Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 3], [0, 5], [1, 2], [1, 4], [2, 3], [3, 4], [4, 5]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Supplied colors=[0, 1, 2, 1, 0, 1] in vertex order. Color labels are nonnegative integer palette indices; retain their actual labels. Properness checks every edge, and using palette [0,k) is separate from merely using at most k distinct labels. Compare k=4. Give (k, candidate_k_palette_valid, chromatic_number, k_equals_chromatic_number, boundary).

k_equals_chromatic_number compares k with the true chromatic number; it does not certify the supplied coloring. Check candidate_k_palette_valid separately.

View answer and explanation(4, Yes, 3, No, k_colorable_means_at_most_k_and_greedy_not_optimality_proof)

Keep the supplied palette validity flag separate from the exhaustive chromatic number and the comparison k=chi. The comparison may be Yes even when the supplied coloring fails. A greedy or arbitrary valid palette alone is not a lower-bound certificate. The exact result, retaining the requested order, is (4, Yes, 3, No, k_colorable_means_at_most_k_and_greedy_not_optimality_proof).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r01-chromatic-number-0087/)

Chromatic number · qf-mod37-chromatic-minimum-boundary · candidate





## 88. Equality of k with the Minimum Is Not Validity of This Coloring

Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 5], [1, 2], [1, 3], [1, 4], [2, 3], [3, 4], [4, 5]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Supplied colors=[0, 1, 0, 2, 0, 1] in vertex order. Color labels are nonnegative integer palette indices; retain their actual labels. Properness checks every edge, and using palette [0,k) is separate from merely using at most k distinct labels. Compare k=4. Give (k, candidate_k_palette_valid, chromatic_number, k_equals_chromatic_number, boundary).

k_equals_chromatic_number compares k with the true chromatic number; it does not certify the supplied coloring. Check candidate_k_palette_valid separately.

View answer and explanation(4, Yes, 3, No, k_colorable_means_at_most_k_and_greedy_not_optimality_proof)

Keep the supplied palette validity flag separate from the exhaustive chromatic number and the comparison k=chi. The comparison may be Yes even when the supplied coloring fails. A greedy or arbitrary valid palette alone is not a lower-bound certificate. The exact result, retaining the requested order, is (4, Yes, 3, No, k_colorable_means_at_most_k_and_greedy_not_optimality_proof).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r01-chromatic-number-0088/)

Chromatic number · qf-mod37-chromatic-minimum-boundary · candidate





## 89. Equality of k with the Minimum Is Not Validity of This Coloring

Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 5], [1, 2], [1, 3], [1, 4], [2, 3], [3, 4], [4, 5]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Supplied colors=[0, 1, 2, 0, 2, 1] in vertex order. Color labels are nonnegative integer palette indices; retain their actual labels. Properness checks every edge, and using palette [0,k) is separate from merely using at most k distinct labels. Compare k=4. Give (k, candidate_k_palette_valid, chromatic_number, k_equals_chromatic_number, boundary).

k_equals_chromatic_number compares k with the true chromatic number; it does not certify the supplied coloring. Check candidate_k_palette_valid separately.

View answer and explanation(4, Yes, 3, No, k_colorable_means_at_most_k_and_greedy_not_optimality_proof)

Keep the supplied palette validity flag separate from the exhaustive chromatic number and the comparison k=chi. The comparison may be Yes even when the supplied coloring fails. A greedy or arbitrary valid palette alone is not a lower-bound certificate. The exact result, retaining the requested order, is (4, Yes, 3, No, k_colorable_means_at_most_k_and_greedy_not_optimality_proof).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r01-chromatic-number-0089/)

Chromatic number · qf-mod37-chromatic-minimum-boundary · candidate





## 90. Equality of k with the Minimum Is Not Validity of This Coloring

Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 3], [0, 5], [1, 2], [1, 3], [1, 4], [2, 3], [3, 4], [4, 5]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Supplied colors=[0, 1, 0, 2, 0, 1] in vertex order. Color labels are nonnegative integer palette indices; retain their actual labels. Properness checks every edge, and using palette [0,k) is separate from merely using at most k distinct labels. Compare k=4. Give (k, candidate_k_palette_valid, chromatic_number, k_equals_chromatic_number, boundary).

k_equals_chromatic_number compares k with the true chromatic number; it does not certify the supplied coloring. Check candidate_k_palette_valid separately.

View answer and explanation(4, Yes, 3, No, k_colorable_means_at_most_k_and_greedy_not_optimality_proof)

Keep the supplied palette validity flag separate from the exhaustive chromatic number and the comparison k=chi. The comparison may be Yes even when the supplied coloring fails. A greedy or arbitrary valid palette alone is not a lower-bound certificate. The exact result, retaining the requested order, is (4, Yes, 3, No, k_colorable_means_at_most_k_and_greedy_not_optimality_proof).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r01-chromatic-number-0090/)

Chromatic number · qf-mod37-chromatic-minimum-boundary · candidate





## 91. Equality of k with the Minimum Is Not Validity of This Coloring

Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 3], [0, 5], [1, 2], [1, 3], [1, 4], [2, 3], [3, 4], [4, 5]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Supplied colors=[0, 1, 2, 3, 0, 1] in vertex order. Color labels are nonnegative integer palette indices; retain their actual labels. Properness checks every edge, and using palette [0,k) is separate from merely using at most k distinct labels. Compare k=5. Give (k, candidate_k_palette_valid, chromatic_number, k_equals_chromatic_number, boundary).

k_equals_chromatic_number compares k with the true chromatic number; it does not certify the supplied coloring. Check candidate_k_palette_valid separately.

View answer and explanation(5, Yes, 4, No, k_colorable_means_at_most_k_and_greedy_not_optimality_proof)

Keep the supplied palette validity flag separate from the exhaustive chromatic number and the comparison k=chi. The comparison may be Yes even when the supplied coloring fails. A greedy or arbitrary valid palette alone is not a lower-bound certificate. The exact result, retaining the requested order, is (5, Yes, 4, No, k_colorable_means_at_most_k_and_greedy_not_optimality_proof).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r01-chromatic-number-0091/)

Chromatic number · qf-mod37-chromatic-minimum-boundary · candidate





## 92. Equality of k with the Minimum Is Not Validity of This Coloring

Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 5], [1, 2], [2, 3], [2, 5], [3, 4], [4, 5]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Supplied colors=[0, 1, 0, 1, 0, 1] in vertex order. Color labels are nonnegative integer palette indices; retain their actual labels. Properness checks every edge, and using palette [0,k) is separate from merely using at most k distinct labels. Compare k=3. Give (k, candidate_k_palette_valid, chromatic_number, k_equals_chromatic_number, boundary).

k_equals_chromatic_number compares k with the true chromatic number; it does not certify the supplied coloring. Check candidate_k_palette_valid separately.

View answer and explanation(3, Yes, 2, No, k_colorable_means_at_most_k_and_greedy_not_optimality_proof)

Keep the supplied palette validity flag separate from the exhaustive chromatic number and the comparison k=chi. The comparison may be Yes even when the supplied coloring fails. A greedy or arbitrary valid palette alone is not a lower-bound certificate. The exact result, retaining the requested order, is (3, Yes, 2, No, k_colorable_means_at_most_k_and_greedy_not_optimality_proof).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r01-chromatic-number-0092/)

Chromatic number · qf-mod37-chromatic-minimum-boundary · candidate





## 93. Equality of k with the Minimum Is Not Validity of This Coloring

Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 5], [1, 2], [2, 3], [2, 5], [3, 4], [4, 5]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Supplied colors=[0, 1, 2, 0, 2, 1] in vertex order. Color labels are nonnegative integer palette indices; retain their actual labels. Properness checks every edge, and using palette [0,k) is separate from merely using at most k distinct labels. Compare k=4. Give (k, candidate_k_palette_valid, chromatic_number, k_equals_chromatic_number, boundary).

k_equals_chromatic_number compares k with the true chromatic number; it does not certify the supplied coloring. Check candidate_k_palette_valid separately.

View answer and explanation(4, Yes, 3, No, k_colorable_means_at_most_k_and_greedy_not_optimality_proof)

Keep the supplied palette validity flag separate from the exhaustive chromatic number and the comparison k=chi. The comparison may be Yes even when the supplied coloring fails. A greedy or arbitrary valid palette alone is not a lower-bound certificate. The exact result, retaining the requested order, is (4, Yes, 3, No, k_colorable_means_at_most_k_and_greedy_not_optimality_proof).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r01-chromatic-number-0093/)

Chromatic number · qf-mod37-chromatic-minimum-boundary · candidate





## 94. Equality of k with the Minimum Is Not Validity of This Coloring

Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 3], [0, 5], [1, 2], [2, 3], [2, 5], [3, 4], [4, 5]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Supplied colors=[0, 1, 0, 1, 0, 1] in vertex order. Color labels are nonnegative integer palette indices; retain their actual labels. Properness checks every edge, and using palette [0,k) is separate from merely using at most k distinct labels. Compare k=3. Give (k, candidate_k_palette_valid, chromatic_number, k_equals_chromatic_number, boundary).

k_equals_chromatic_number compares k with the true chromatic number; it does not certify the supplied coloring. Check candidate_k_palette_valid separately.

View answer and explanation(3, Yes, 2, No, k_colorable_means_at_most_k_and_greedy_not_optimality_proof)

Keep the supplied palette validity flag separate from the exhaustive chromatic number and the comparison k=chi. The comparison may be Yes even when the supplied coloring fails. A greedy or arbitrary valid palette alone is not a lower-bound certificate. The exact result, retaining the requested order, is (3, Yes, 2, No, k_colorable_means_at_most_k_and_greedy_not_optimality_proof).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r01-chromatic-number-0094/)

Chromatic number · qf-mod37-chromatic-minimum-boundary · candidate





## 95. Equality of k with the Minimum Is Not Validity of This Coloring

Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 3], [0, 5], [1, 2], [2, 3], [2, 5], [3, 4], [4, 5]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Supplied colors=[0, 1, 2, 1, 0, 1] in vertex order. Color labels are nonnegative integer palette indices; retain their actual labels. Properness checks every edge, and using palette [0,k) is separate from merely using at most k distinct labels. Compare k=4. Give (k, candidate_k_palette_valid, chromatic_number, k_equals_chromatic_number, boundary).

k_equals_chromatic_number compares k with the true chromatic number; it does not certify the supplied coloring. Check candidate_k_palette_valid separately.

View answer and explanation(4, Yes, 3, No, k_colorable_means_at_most_k_and_greedy_not_optimality_proof)

Keep the supplied palette validity flag separate from the exhaustive chromatic number and the comparison k=chi. The comparison may be Yes even when the supplied coloring fails. A greedy or arbitrary valid palette alone is not a lower-bound certificate. The exact result, retaining the requested order, is (4, Yes, 3, No, k_colorable_means_at_most_k_and_greedy_not_optimality_proof).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r01-chromatic-number-0095/)

Chromatic number · qf-mod37-chromatic-minimum-boundary · candidate





## 96. Equality of k with the Minimum Is Not Validity of This Coloring

Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 5], [1, 2], [1, 3], [2, 3], [2, 5], [3, 4], [4, 5]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Supplied colors=[0, 1, 0, 2, 0, 1] in vertex order. Color labels are nonnegative integer palette indices; retain their actual labels. Properness checks every edge, and using palette [0,k) is separate from merely using at most k distinct labels. Compare k=4. Give (k, candidate_k_palette_valid, chromatic_number, k_equals_chromatic_number, boundary).

k_equals_chromatic_number compares k with the true chromatic number; it does not certify the supplied coloring. Check candidate_k_palette_valid separately.

View answer and explanation(4, Yes, 3, No, k_colorable_means_at_most_k_and_greedy_not_optimality_proof)

Keep the supplied palette validity flag separate from the exhaustive chromatic number and the comparison k=chi. The comparison may be Yes even when the supplied coloring fails. A greedy or arbitrary valid palette alone is not a lower-bound certificate. The exact result, retaining the requested order, is (4, Yes, 3, No, k_colorable_means_at_most_k_and_greedy_not_optimality_proof).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r01-chromatic-number-0096/)

Chromatic number · qf-mod37-chromatic-minimum-boundary · candidate





## 97. Equality of k with the Minimum Is Not Validity of This Coloring

Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 5], [1, 2], [1, 3], [2, 3], [2, 5], [3, 4], [4, 5]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Supplied colors=[0, 1, 2, 0, 2, 1] in vertex order. Color labels are nonnegative integer palette indices; retain their actual labels. Properness checks every edge, and using palette [0,k) is separate from merely using at most k distinct labels. Compare k=4. Give (k, candidate_k_palette_valid, chromatic_number, k_equals_chromatic_number, boundary).

k_equals_chromatic_number compares k with the true chromatic number; it does not certify the supplied coloring. Check candidate_k_palette_valid separately.

View answer and explanation(4, Yes, 3, No, k_colorable_means_at_most_k_and_greedy_not_optimality_proof)

Keep the supplied palette validity flag separate from the exhaustive chromatic number and the comparison k=chi. The comparison may be Yes even when the supplied coloring fails. A greedy or arbitrary valid palette alone is not a lower-bound certificate. The exact result, retaining the requested order, is (4, Yes, 3, No, k_colorable_means_at_most_k_and_greedy_not_optimality_proof).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r01-chromatic-number-0097/)

Chromatic number · qf-mod37-chromatic-minimum-boundary · candidate





## 98. Equality of k with the Minimum Is Not Validity of This Coloring

Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 3], [0, 5], [1, 2], [1, 3], [2, 3], [2, 5], [3, 4], [4, 5]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Supplied colors=[0, 1, 0, 2, 0, 1] in vertex order. Color labels are nonnegative integer palette indices; retain their actual labels. Properness checks every edge, and using palette [0,k) is separate from merely using at most k distinct labels. Compare k=4. Give (k, candidate_k_palette_valid, chromatic_number, k_equals_chromatic_number, boundary).

k_equals_chromatic_number compares k with the true chromatic number; it does not certify the supplied coloring. Check candidate_k_palette_valid separately.

View answer and explanation(4, Yes, 3, No, k_colorable_means_at_most_k_and_greedy_not_optimality_proof)

Keep the supplied palette validity flag separate from the exhaustive chromatic number and the comparison k=chi. The comparison may be Yes even when the supplied coloring fails. A greedy or arbitrary valid palette alone is not a lower-bound certificate. The exact result, retaining the requested order, is (4, Yes, 3, No, k_colorable_means_at_most_k_and_greedy_not_optimality_proof).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r01-chromatic-number-0098/)

Chromatic number · qf-mod37-chromatic-minimum-boundary · candidate





## 99. Equality of k with the Minimum Is Not Validity of This Coloring

Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 3], [0, 5], [1, 2], [1, 3], [2, 3], [2, 5], [3, 4], [4, 5]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Supplied colors=[0, 1, 2, 3, 0, 1] in vertex order. Color labels are nonnegative integer palette indices; retain their actual labels. Properness checks every edge, and using palette [0,k) is separate from merely using at most k distinct labels. Compare k=5. Give (k, candidate_k_palette_valid, chromatic_number, k_equals_chromatic_number, boundary).

k_equals_chromatic_number compares k with the true chromatic number; it does not certify the supplied coloring. Check candidate_k_palette_valid separately.

View answer and explanation(5, Yes, 4, No, k_colorable_means_at_most_k_and_greedy_not_optimality_proof)

Keep the supplied palette validity flag separate from the exhaustive chromatic number and the comparison k=chi. The comparison may be Yes even when the supplied coloring fails. A greedy or arbitrary valid palette alone is not a lower-bound certificate. The exact result, retaining the requested order, is (5, Yes, 4, No, k_colorable_means_at_most_k_and_greedy_not_optimality_proof).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r01-chromatic-number-0099/)

Chromatic number · qf-mod37-chromatic-minimum-boundary · candidate





## 100. Equality of k with the Minimum Is Not Validity of This Coloring

Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 5], [1, 2], [1, 4], [2, 3], [2, 5], [3, 4], [4, 5]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Supplied colors=[0, 1, 0, 1, 0, 1] in vertex order. Color labels are nonnegative integer palette indices; retain their actual labels. Properness checks every edge, and using palette [0,k) is separate from merely using at most k distinct labels. Compare k=3. Give (k, candidate_k_palette_valid, chromatic_number, k_equals_chromatic_number, boundary).

k_equals_chromatic_number compares k with the true chromatic number; it does not certify the supplied coloring. Check candidate_k_palette_valid separately.

View answer and explanation(3, Yes, 2, No, k_colorable_means_at_most_k_and_greedy_not_optimality_proof)

Keep the supplied palette validity flag separate from the exhaustive chromatic number and the comparison k=chi. The comparison may be Yes even when the supplied coloring fails. A greedy or arbitrary valid palette alone is not a lower-bound certificate. The exact result, retaining the requested order, is (3, Yes, 2, No, k_colorable_means_at_most_k_and_greedy_not_optimality_proof).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r01-chromatic-number-0100/)

Chromatic number · qf-mod37-chromatic-minimum-boundary · candidate



[Canonical question JSONL](https://ocme.evemisslab.com/data/questions/batches/batch-20261014-advance-20261106-r01-chromatic-number-foundations/questions.jsonl) · [English JSONL](https://ocme.evemisslab.com/data/i18n/questions/batch-20261014-advance-20261106-r01-chromatic-number-foundations.en.jsonl) · [Validation report](https://ocme.evemisslab.com/data/questions/batches/batch-20261014-advance-20261106-r01-chromatic-number-foundations/validation-report.json)