# Question 19426: Equality of k with the Minimum Is Not Validity of This Coloring

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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.

R0037-19426 · q-20261014-advance-20261106-r01-chromatic-number-0076



## Original question and givens

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).


- G001 — 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).





## 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-minimum-boundary;original snapshot and mathematics retained

After

explicit ordered source output fields=[k, candidate_k_palette_valid, chromatic_number, k_equals_chromatic_number, 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, 5], [1, 2], [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, 5];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, S003





S005 · impliesBefore

original edge index=2;actual endpoint vertices=[1, 2];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, S004





S006 · impliesBefore

original edge index=3;actual endpoint vertices=[2, 3];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, S005





S007 · impliesBefore

original edge index=4;actual endpoint vertices=[3, 4];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=[4, 5];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, S007





S009 · calculationBefore

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

After

chromatic distinct used label count = 2

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, S008





S010 · impliesBefore

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

After

increasingly ordered DISTINCT used labels=[0, 1];distinct count=2;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, S009





S011 · impliesBefore

requested ACTUAL palette[0,3);ORIGINAL labels=[0, 1, 0, 1, 0, 1];per-vertex label<k flags=[Yes, Yes, Yes, Yes, Yes, Yes];proper=Yes

After

within requested palette=Yes;THIS candidate upper certificate=Yes;distinct used-label count=2 is a DIFFERENT test

Conditions: none

Even if at most k distinct labels are used,the supplied palette certificate fails when original labels lie outside[0,k).Semantic relabeling must not replace source data.Failure rejects this candidate,not existence of another k-coloring.

Step identifiers and dependenciesRule: chromatic_original_palette_range_not_semantic_renaming · Uses: G001, S010





S012 · impliesBefore

ACTUAL graph Fin(6);try palette[0,1);original vertex order0..5

After

search ALL earlier lexicographic prefixes:vertex order fixed,each color tried increasingly;only an actual earlier-edge conflict may prune

Conditions: none

Search each k completely rather than treating a failed greedy run as non-colorability.Fixed original vertex order and increasing colors define the source's first lexicographic witness.Prune only an actual edge conflict;otherwise continue or exhaustively backtrack.

Step identifiers and dependenciesRule: chromatic_complete_lex_search_not_greedy_optimality · Uses: G001, S011





S013 · impliesBefore

palette=1;original vertex=0;assigned ORIGINAL prefix=[];try next ascending color=0

After

ALL earlier adjacent vertices already using this label=[];admissible extension=Yes

Conditions: none

List all actual earlier-neighbor conflicts for this prefix/color.A conflict rejects every completion of that extension;otherwise recursively check all remaining vertices rather than calling a local pass a complete solution.

Step identifiers and dependenciesRule: chromatic_every_prefix_color_conflict_or_recursion · Uses: G001, S012





S014 · impliesBefore

palette=1;original vertex=1;assigned ORIGINAL prefix=[0];try next ascending color=0

After

ALL earlier adjacent vertices already using this label=[0];admissible extension=No

Conditions: none

List all actual earlier-neighbor conflicts for this prefix/color.A conflict rejects every completion of that extension;otherwise recursively check all remaining vertices rather than calling a local pass a complete solution.

Step identifiers and dependenciesRule: chromatic_every_prefix_color_conflict_or_recursion · Uses: G001, S013





S015 · impliesBefore

palette=1;original vertex=1;original fixed prefix=[0];ALL colors0..0 attempted

After

no valid completion of this prefix in this palette

Conditions: none

All allowed labels at this vertex have been tried,so this fixed prefix has no completion.Only this conclusion at the root excludes the whole k-palette.

Step identifiers and dependenciesRule: chromatic_full_palette_exhaustion_at_prefix · Uses: G001, S014





S016 · impliesBefore

palette=1;original prefix=[];vertex=0;chosen color=0;ALL its later completions exhausted

After

restore this unassigned coordinate and try the next ascending color

Conditions: none

Backtrack only after the whole subtree fails.Every later completion of this branch was excluded by an actual conflict or a completely exhausted subtree,not one sampled attempt.

Step identifiers and dependenciesRule: chromatic_exhausted_branch_backtracking_not_one_bad_sample · Uses: G001, S015





S017 · impliesBefore

palette=1;original vertex=0;original fixed prefix=[];ALL colors0..0 attempted

After

no valid completion of this prefix in this palette

Conditions: none

All allowed labels at this vertex have been tried,so this fixed prefix has no completion.Only this conclusion at the root excludes the whole k-palette.

Step identifiers and dependenciesRule: chromatic_full_palette_exhaustion_at_prefix · Uses: G001, S016





S018 · impliesBefore

palette[0,1) searched from EMPTY root prefix with full branch exclusions

After

NO proper coloring exists in this palette;any smaller palette is a subset of it

Conditions: none

Exhausting the complete root tree supplies the lower certificate.Any smaller palette embeds in this one and is excluded as well;no greedy path establishes this conclusion.

Step identifiers and dependenciesRule: chromatic_actual_lower_palette_exhaustion · Uses: G001, S017





S019 · impliesBefore

ACTUAL graph Fin(6);try palette[0,2);original vertex order0..5

After

search ALL earlier lexicographic prefixes:vertex order fixed,each color tried increasingly;only an actual earlier-edge conflict may prune

Conditions: none

Search each k completely rather than treating a failed greedy run as non-colorability.Fixed original vertex order and increasing colors define the source's first lexicographic witness.Prune only an actual edge conflict;otherwise continue or exhaustively backtrack.

Step identifiers and dependenciesRule: chromatic_complete_lex_search_not_greedy_optimality · Uses: G001, S018





S020 · impliesBefore

palette=2;original vertex=0;assigned ORIGINAL prefix=[];try next ascending color=0

After

ALL earlier adjacent vertices already using this label=[];admissible extension=Yes

Conditions: none

List all actual earlier-neighbor conflicts for this prefix/color.A conflict rejects every completion of that extension;otherwise recursively check all remaining vertices rather than calling a local pass a complete solution.

Step identifiers and dependenciesRule: chromatic_every_prefix_color_conflict_or_recursion · Uses: G001, S019





S021 · impliesBefore

palette=2;original vertex=1;assigned ORIGINAL prefix=[0];try next ascending color=0

After

ALL earlier adjacent vertices already using this label=[0];admissible extension=No

Conditions: none

List all actual earlier-neighbor conflicts for this prefix/color.A conflict rejects every completion of that extension;otherwise recursively check all remaining vertices rather than calling a local pass a complete solution.

Step identifiers and dependenciesRule: chromatic_every_prefix_color_conflict_or_recursion · Uses: G001, S020





S022 · impliesBefore

palette=2;original vertex=1;assigned ORIGINAL prefix=[0];try next ascending color=1

After

ALL earlier adjacent vertices already using this label=[];admissible extension=Yes

Conditions: none

List all actual earlier-neighbor conflicts for this prefix/color.A conflict rejects every completion of that extension;otherwise recursively check all remaining vertices rather than calling a local pass a complete solution.

Step identifiers and dependenciesRule: chromatic_every_prefix_color_conflict_or_recursion · Uses: G001, S021





S023 · impliesBefore

palette=2;original vertex=2;assigned ORIGINAL prefix=[0, 1];try next ascending color=0

After

ALL earlier adjacent vertices already using this label=[];admissible extension=Yes

Conditions: none

List all actual earlier-neighbor conflicts for this prefix/color.A conflict rejects every completion of that extension;otherwise recursively check all remaining vertices rather than calling a local pass a complete solution.

Step identifiers and dependenciesRule: chromatic_every_prefix_color_conflict_or_recursion · Uses: G001, S022





S024 · impliesBefore

palette=2;original vertex=3;assigned ORIGINAL prefix=[0, 1, 0];try next ascending color=0

After

ALL earlier adjacent vertices already using this label=[2];admissible extension=No

Conditions: none

List all actual earlier-neighbor conflicts for this prefix/color.A conflict rejects every completion of that extension;otherwise recursively check all remaining vertices rather than calling a local pass a complete solution.

Step identifiers and dependenciesRule: chromatic_every_prefix_color_conflict_or_recursion · Uses: G001, S023





S025 · impliesBefore

palette=2;original vertex=3;assigned ORIGINAL prefix=[0, 1, 0];try next ascending color=1

After

ALL earlier adjacent vertices already using this label=[];admissible extension=Yes

Conditions: none

List all actual earlier-neighbor conflicts for this prefix/color.A conflict rejects every completion of that extension;otherwise recursively check all remaining vertices rather than calling a local pass a complete solution.

Step identifiers and dependenciesRule: chromatic_every_prefix_color_conflict_or_recursion · Uses: G001, S024





S026 · impliesBefore

palette=2;original vertex=4;assigned ORIGINAL prefix=[0, 1, 0, 1];try next ascending color=0

After

ALL earlier adjacent vertices already using this label=[];admissible extension=Yes

Conditions: none

List all actual earlier-neighbor conflicts for this prefix/color.A conflict rejects every completion of that extension;otherwise recursively check all remaining vertices rather than calling a local pass a complete solution.

Step identifiers and dependenciesRule: chromatic_every_prefix_color_conflict_or_recursion · Uses: G001, S025





S027 · impliesBefore

palette=2;original vertex=5;assigned ORIGINAL prefix=[0, 1, 0, 1, 0];try next ascending color=0

After

ALL earlier adjacent vertices already using this label=[0, 4];admissible extension=No

Conditions: none

List all actual earlier-neighbor conflicts for this prefix/color.A conflict rejects every completion of that extension;otherwise recursively check all remaining vertices rather than calling a local pass a complete solution.

Step identifiers and dependenciesRule: chromatic_every_prefix_color_conflict_or_recursion · Uses: G001, S026





S028 · impliesBefore

palette=2;original vertex=5;assigned ORIGINAL prefix=[0, 1, 0, 1, 0];try next ascending color=1

After

ALL earlier adjacent vertices already using this label=[];admissible extension=Yes

Conditions: none

List all actual earlier-neighbor conflicts for this prefix/color.A conflict rejects every completion of that extension;otherwise recursively check all remaining vertices rather than calling a local pass a complete solution.

Step identifiers and dependenciesRule: chromatic_every_prefix_color_conflict_or_recursion · Uses: G001, S027





S029 · impliesBefore

FIRST completed vertex-order increasing-color search witness=[0, 1, 0, 1, 0, 1];palette size=2;ALL smaller palettes exhausted

After

every ORIGINAL edge endpoints different;constructive chi<= 2 and exhaustive chi>= 2;chi=2;first LEX witness retained

Conditions: none

The complete proper original-edge coloring gives the upper bound;the preceding exhaustive failures give the lower bound.Their equality establishes the exact minimum.Fixed DFS order preserves the source's first lexicographic witness,not an arbitrary optimum.A nonempty edgeless graph succeeds first at k=1 with all zeros;no fictitious0-color search is needed.

Step identifiers and dependenciesRule: chromatic_constructive_upper_and_exhausted_lower_tie_order · Uses: G001, S028





S030 · impliesBefore

requested k=3;THIS supplied candidate validity=Yes;SEPARATE exact chi=2;SEPARATE canonical optimum witness=[0, 1, 0, 1, 0, 1]

After

k equals chromatic number=No;comparison does NOT certify supplied colors;at-most-k and exact-k-minimum are separate

Conditions: none

k may equal the true chromatic number while the supplied coloring fails.Or a valid k-palette may only give chi≤k with equality false.Keep these separate questions and do not copy one successful flag into another.

Step identifiers and dependenciesRule: chromatic_k_equals_chi_independent_of_supplied_candidate · Uses: G001, S029









## Reverse checks


- S030 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S029 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S028 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S027 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S026 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S025 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S024 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S023 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S022 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S021 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S020 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S019 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S018 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S017 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S016 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S015 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S014 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S013 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- 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 — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- 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 — Recompute chromatic distinct used label count with exact rational arithmetic, not rounded decimals.
- 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

(3, Yes, 2, No, k_colorable_means_at_most_k_and_greedy_not_optimality_proof)

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: de4ff9713787f1462907a8ed52c5b116df000d80c040ccdc64170f36aa700e0f
- candidate record SHA-256: ea5543f25a707f1b698be1efcdebe846cf0b2bb453a9cc5331fa749f7f82523f
- 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-0076.json)