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