# Question 15687: A Nonoptimal Valid Coloring Still Supplies an Upper Bound [Return to catalogue page 79](https://ocme.evemisslab.com/en/questions/page/79/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261014-advance-20261031-r05-graph-coloring-foundations/#q-20261014-advance-20261031-r05-graph-coloring-0062) 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. R0030-15687 · q-20261014-advance-20261031-r05-graph-coloring-0062 ## Original question and givens Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5]]. Normalize each edge increasingly, then sort edges lexicographically; reversed duplicates and loops are forbidden. Keep isolated vertices in the carrier. No directed-graph conclusion is inferred. Total vertex colors=[11, 12, 11, 12, 11, 12] are labels, not cardinalities. Only adjacent endpoints must differ; nonadjacent vertices may share a color. A proper k-coloring uses at most k distinct labels, even if their numerical names exceed k−1. One such coloring gives only a chromatic-number upper bound, not optimality or a general greedy guarantee. Proposed k=4. Give (used_count,k,proper?,certificate?,conclusion,upper_bound_not_exact_optimum). A valid certificate yields chromatic_number_at_most_k with the supplied numeric k; otherwise no_upper_bound_from_this_coloring. - G001 — Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5]]. Normalize each edge increasingly, then sort edges lexicographically; reversed duplicates and loops are forbidden. Keep isolated vertices in the carrier. No directed-graph conclusion is inferred. Total vertex colors=[11, 12, 11, 12, 11, 12] are labels, not cardinalities. Only adjacent endpoints must differ; nonadjacent vertices may share a color. A proper k-coloring uses at most k distinct labels, even if their numerical names exceed k−1. One such coloring gives only a chromatic-number upper bound, not optimality or a general greedy guarantee. Proposed k=4. Give (used_count,k,proper?,certificate?,conclusion,upper_bound_not_exact_optimum). A valid certificate yields chromatic_number_at_most_k with the supplied numeric k; otherwise no_upper_bound_from_this_coloring. ## 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-graph-coloring](https://ocme.evemisslab.com/en/mko/mko-graph-coloring/) ## Step-by-step derivation S001 · impliesBefore complete undirected vertex carrier Fin(6);original edges=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5]] After normalized sorted simple edges=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5]];all vertices including isolates=[0, 1, 2, 3, 4, 5] Conditions: none Normalize endpoints and then sort edges lexicographically.Reversed duplicates are still duplicates and are rejected,as are loops.The complete carrier includes vertices absent from the edge list. Step identifiers and dependenciesRule: graph_complete_simple_carrier_normalized_edge_order · Uses: G001 S002 · impliesBefore colors:normalized edge index=0;edge=[0, 1];endpoint labels=[11, 12] After different endpoints=Yes;same-label conflict=No Conditions: none Check the endpoints of each actual edge.Reused labels at nonadjacent vertices are not conflicts,and one crossing edge does not certify all other edges. Step identifiers and dependenciesRule: graph_every_edge_endpoint_constraint · Uses: G001, S001 S003 · impliesBefore colors:normalized edge index=1;edge=[1, 2];endpoint labels=[12, 11] After different endpoints=Yes;same-label conflict=No Conditions: none Check the endpoints of each actual edge.Reused labels at nonadjacent vertices are not conflicts,and one crossing edge does not certify all other edges. Step identifiers and dependenciesRule: graph_every_edge_endpoint_constraint · Uses: G001, S002 S004 · impliesBefore colors:normalized edge index=2;edge=[2, 3];endpoint labels=[11, 12] After different endpoints=Yes;same-label conflict=No Conditions: none Check the endpoints of each actual edge.Reused labels at nonadjacent vertices are not conflicts,and one crossing edge does not certify all other edges. Step identifiers and dependenciesRule: graph_every_edge_endpoint_constraint · Uses: G001, S003 S005 · impliesBefore colors:normalized edge index=3;edge=[3, 4];endpoint labels=[12, 11] After different endpoints=Yes;same-label conflict=No Conditions: none Check the endpoints of each actual edge.Reused labels at nonadjacent vertices are not conflicts,and one crossing edge does not certify all other edges. Step identifiers and dependenciesRule: graph_every_edge_endpoint_constraint · Uses: G001, S004 S006 · impliesBefore colors:normalized edge index=4;edge=[4, 5];endpoint labels=[11, 12] After different endpoints=Yes;same-label conflict=No Conditions: none Check the endpoints of each actual edge.Reused labels at nonadjacent vertices are not conflicts,and one crossing edge does not certify all other edges. Step identifiers and dependenciesRule: graph_every_edge_endpoint_constraint · Uses: G001, S005 S007 · impliesBefore total original vertex colors=[11, 12, 11, 12, 11, 12];all normalized conflicts=[] After increasing distinct used labels=[11, 12];used count=2;all edges proper=Yes Conditions: none Colors are names:count distinct used labels rather than maximum label plus1.Only adjacent equal colors conflict;nonadjacent color reuse is allowed. Step identifiers and dependenciesRule: coloring_distinct_label_count_not_numeric_magnitude · Uses: G001, S006 S008 · impliesBefore all-edge proper=Yes;distinct used count=2;original k=4 After at most k=Yes;proper k-coloring certificate=Yes;upper_bound_not_exact_optimum Conditions: none Both edge properness and at most k used colors are required for χ(G)≤k.Rename the used labels into a k-color palette if needed;large numeric names do not invalidate it.Without excluding all smaller k,this is not an optimum certificate. Step identifiers and dependenciesRule: coloring_k_bound_conjunction_not_minimum_chromatic_number · Uses: G001, S007 ## Reverse checks - 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 (2, 4, Yes, Yes, chromatic_number_at_most_4, upper_bound_not_exact_optimum) 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: 159c4dec178968f44867923a74da41aa5d15e999cc4c67c34fa7a2de814727e3 - candidate record SHA-256: 73d301cb56c42c746284392cdd25f196a8925d7a5bc183c0dc5f10a402f8b7be - semantic review receipt SHA-256: 73bc08066ffb1a81eacb2d4e288d03f522aeefd3949fa0d97da41b484ed737bc - question-first witness SHA-256: ac120452a208b257b8270f38351a09dcd832cdbc8d9db6b69933f83e43cc9336 - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261014-advance-20261031-r05-graph-coloring-0062.json)