Question 19454: Proper Coloring Alone Does Not Prove the Chromatic Minimum

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, 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).

• G001 — 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).

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

Step-by-step derivation

source operation=chromatic-proper-audit;original snapshot and mathematics retained

explicit ordered source output fields=[colors, used_colors, color_count, proper, boundary]

ACTUAL complete finite carrier Fin(6);ALL original canonical simple undirected edges=[[0, 1], [0, 2], [0, 3], [0, 5], [1, 2], [2, 3], [3, 4], [4, 5]]

use full original vertex/edge data;exact search domain n<=9;empty carrier and isolated vertices RETAINED

original edge index=0;actual endpoint vertices=[0, 1];ORIGINAL palette labels=[0, 1]

endpoint labels different=Yes

original edge index=1;actual endpoint vertices=[0, 2];ORIGINAL palette labels=[0, 2]

endpoint labels different=Yes

original edge index=2;actual endpoint vertices=[0, 3];ORIGINAL palette labels=[0, 1]

endpoint labels different=Yes

original edge index=3;actual endpoint vertices=[0, 5];ORIGINAL palette labels=[0, 1]

endpoint labels different=Yes

original edge index=4;actual endpoint vertices=[1, 2];ORIGINAL palette labels=[1, 2]

endpoint labels different=Yes

original edge index=5;actual endpoint vertices=[2, 3];ORIGINAL palette labels=[2, 1]

endpoint labels different=Yes

original edge index=6;actual endpoint vertices=[3, 4];ORIGINAL palette labels=[1, 0]

endpoint labels different=Yes

original edge index=7;actual endpoint vertices=[4, 5];ORIGINAL palette labels=[0, 1]

endpoint labels different=Yes

chromatic distinct used label count := sum(1, 1, 1)

chromatic distinct used label count = 3

ORIGINAL complete color array=[0, 1, 2, 1, 0, 1];ALL edge properness flags=[Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes]

increasingly ordered DISTINCT used labels=[0, 1, 2];distinct count=3;proper=Yes;not max_label+1

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, 1, 0, 1], [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.