# Question 19456: Proper Coloring Alone Does Not Prove the Chromatic Minimum [Return to catalogue page 98](https://ocme.evemisslab.com/en/questions/page/98/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261014-advance-20261106-r01-chromatic-number-foundations/#q-20261014-advance-20261106-r01-chromatic-number-0006) This teaching derivation has second-context AI mathematical and bilingual semantic review, limited to the model and assumptions stated in this question. It has no human review or new Lean proof and does not promote the original question or MKO to Canonical truth. R0037-19456 · q-20261014-advance-20261106-r01-chromatic-number-0006 ## Original question and givens 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). - G001 — 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). ## Definition and scope Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate. - [mko-chromatic-number](https://ocme.evemisslab.com/en/mko/mko-chromatic-number/) ## Step-by-step derivation S001 · impliesBefore source operation=chromatic-proper-audit;original snapshot and mathematics retained After explicit ordered source output fields=[colors, used_colors, color_count, proper, boundary] Conditions: none Retain the historical interface marker in the original snapshot while listing ordered source fields here,without JavaScript or external lookup.Field names are not mathematical evidence,and No retains the operation's specific claim scope. Step identifiers and dependenciesRule: source_ordered_interface_explicit_without_javascript · Uses: G001 S002 · impliesBefore ACTUAL complete finite carrier Fin(6);ALL original canonical simple undirected edges=[[0, 1], [0, 2], [0, 5], [1, 2], [1, 3], [2, 3], [3, 4], [4, 5]] After use full original vertex/edge data;exact search domain n<=9;empty carrier and isolated vertices RETAINED Conditions: none Retain every original vertex and edge.This family includes n=0 rather than importing a previous nonempty-carrier guard or deleting isolates.Actual palette labels must not be replaced by semantic relabeling. Step identifiers and dependenciesRule: chromatic_complete_graph_not_semantic_color_relabeling · Uses: G001, S001 S003 · impliesBefore original edge index=0;actual endpoint vertices=[0, 1];ORIGINAL palette labels=[0, 1] After endpoint labels different=Yes Conditions: none Check different colors on every original edge.Numeric labels distinguish colors,not the number of colors.One failed supplied coloring does not establish nonexistence of another proper coloring. Step identifiers and dependenciesRule: chromatic_every_original_edge_properness · Uses: G001, S002 S004 · impliesBefore original edge index=1;actual endpoint vertices=[0, 2];ORIGINAL palette labels=[0, 2] After endpoint labels different=Yes Conditions: none Check different colors on every original edge.Numeric labels distinguish colors,not the number of colors.One failed supplied coloring does not establish nonexistence of another proper coloring. Step identifiers and dependenciesRule: chromatic_every_original_edge_properness · Uses: G001, S003 S005 · impliesBefore original edge index=2;actual endpoint vertices=[0, 5];ORIGINAL palette labels=[0, 2] After endpoint labels different=Yes Conditions: none Check different colors on every original edge.Numeric labels distinguish colors,not the number of colors.One failed supplied coloring does not establish nonexistence of another proper coloring. Step identifiers and dependenciesRule: chromatic_every_original_edge_properness · Uses: G001, S004 S006 · impliesBefore original edge index=3;actual endpoint vertices=[1, 2];ORIGINAL palette labels=[1, 2] After endpoint labels different=Yes Conditions: none Check different colors on every original edge.Numeric labels distinguish colors,not the number of colors.One failed supplied coloring does not establish nonexistence of another proper coloring. Step identifiers and dependenciesRule: chromatic_every_original_edge_properness · Uses: G001, S005 S007 · impliesBefore original edge index=4;actual endpoint vertices=[1, 3];ORIGINAL palette labels=[1, 0] After endpoint labels different=Yes Conditions: none Check different colors on every original edge.Numeric labels distinguish colors,not the number of colors.One failed supplied coloring does not establish nonexistence of another proper coloring. Step identifiers and dependenciesRule: chromatic_every_original_edge_properness · Uses: G001, S006 S008 · impliesBefore original edge index=5;actual endpoint vertices=[2, 3];ORIGINAL palette labels=[2, 0] After endpoint labels different=Yes Conditions: none Check different colors on every original edge.Numeric labels distinguish colors,not the number of colors.One failed supplied coloring does not establish nonexistence of another proper coloring. Step identifiers and dependenciesRule: chromatic_every_original_edge_properness · Uses: G001, S007 S009 · impliesBefore original edge index=6;actual endpoint vertices=[3, 4];ORIGINAL palette labels=[0, 1] After endpoint labels different=Yes Conditions: none Check different colors on every original edge.Numeric labels distinguish colors,not the number of colors.One failed supplied coloring does not establish nonexistence of another proper coloring. Step identifiers and dependenciesRule: chromatic_every_original_edge_properness · Uses: G001, S008 S010 · impliesBefore original edge index=7;actual endpoint vertices=[4, 5];ORIGINAL palette labels=[1, 2] After endpoint labels different=Yes Conditions: none Check different colors on every original edge.Numeric labels distinguish colors,not the number of colors.One failed supplied coloring does not establish nonexistence of another proper coloring. Step identifiers and dependenciesRule: chromatic_every_original_edge_properness · Uses: G001, S009 S011 · calculationBefore chromatic distinct used label count := sum(1, 1, 1) After chromatic distinct used label count = 3 Conditions: none Substitute original roles and established intermediates exactly,retaining parameter,outcome,target and denominator roles without replacing the claim. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S010 S012 · impliesBefore ORIGINAL complete color array=[0, 1, 2, 0, 1, 2];ALL edge properness flags=[Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes] After increasingly ordered DISTINCT used labels=[0, 1, 2];distinct count=3;proper=Yes;not max_label+1 Conditions: none Count distinct labels rather than max_label+1,with0 for an empty array.A proper supplied coloring gives an upper bound,not minimum status from greediness or small labels. Step identifiers and dependenciesRule: chromatic_distinct_label_count_not_palette_range_or_minimum · Uses: G001, S011 ## Reverse checks - S012 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S011 · premise_and_result_audit — Recompute chromatic distinct used label count with exact rational arithmetic, not rounded decimals. - S010 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S009 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S008 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S007 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S006 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S005 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S004 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S003 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S002 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. ## Result ([0, 1, 2, 0, 1, 2], [0, 1, 2], 3, Yes, proper_coloring_only) The full substitutions and premise checks give the original answer; source identity,format and status are unchanged. ## This question review and source binding Review date: 2026-10-26 Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof. Source hashes and review receipts - input commit: 0f7be61e2c22317a4075561dc822c87d881a50db - original question SHA-256: adc28406850c648a9e43fefe089280adaf0c169cf3c8cf8fa23d26fd08386ba4 - candidate record SHA-256: 3c9041cc4fc4175e641539bbae93d199c17f98ce83159e681a6f289a0ac2258d - semantic review receipt SHA-256: 065507b602ceee60cf47a700d387a51940e0c6724c35a3336a7a7948a199ed80 - question-first witness SHA-256: 7203f33b030792559673b0fee520a8c0e70903c4dd970b35293aa40852f333d6 - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261014-advance-20261106-r01-chromatic-number-0006.json)