# Question 15557: An Actual Simple Odd Cycle Certifies Nonbipartiteness [Return to catalogue page 78](https://ocme.evemisslab.com/en/questions/page/78/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261014-advance-20261031-r02-bipartite-graph-foundations/#q-20261014-advance-20261031-r02-bipartite-graph-0057) 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. R0030-15557 · q-20261014-advance-20261031-r02-bipartite-graph-0057 ## Original question and givens Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [0, 2], [4, 5], [3, 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. Ordered cycle vertices=[0, 1, 2], with the closing edge from the last back to the first implicit. Vertices must be distinct, the cycle length odd and at least 3, and every cycle edge present. Give (cycle,cycle_edges,length,certificate?,odd_cycle_proves_nonbipartite). Normalize each edge's endpoints but retain traversal order in cycle_edges. - G001 — Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [0, 2], [4, 5], [3, 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. Ordered cycle vertices=[0, 1, 2], with the closing edge from the last back to the first implicit. Vertices must be distinct, the cycle length odd and at least 3, and every cycle edge present. Give (cycle,cycle_edges,length,certificate?,odd_cycle_proves_nonbipartite). Normalize each edge's endpoints but retain traversal order in cycle_edges. ## 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-bipartite-graph](https://ocme.evemisslab.com/en/mko/mko-bipartite-graph/) ## Step-by-step derivation S001 · impliesBefore complete undirected vertex carrier Fin(6);original edges=[[0, 1], [1, 2], [0, 2], [4, 5], [3, 5]] After normalized sorted simple edges=[[0, 1], [0, 2], [1, 2], [3, 5], [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 ordered distinct cycle vertices=[0, 1, 2];length=3;closing last-first edge is implicit After assume a valid two-part assignment;swap names if needed so first vertex has part0 Conditions: none Assume a valid bipartition and name its first vertex's part0 without loss of generality.Every actual cycle edge must flip parts.The list does not repeat its first vertex;the closing edge is checked separately. Step identifiers and dependenciesRule: odd_cycle_two_part_assumption_not_failed_sample · Uses: G001, S001 S003 · calculationBefore odd cycle edge0 forced next part := sub(1, 0) After odd cycle edge0 forced next part = 1 Conditions: none Compute exactly from original roles and established intermediates,retaining indices,signs,denominators and the operation's carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S002 S004 · impliesBefore cycle traversal index=0;from=0;to=1;normalized actual edge=[0, 1];forced current part=0 After forced next part=1;edge present=Yes Conditions: none Keep traversal order while normalizing each edge.Show both its membership in the complete graph and the forced part flip,including the final return. Step identifiers and dependenciesRule: odd_cycle_every_edge_including_closure · Uses: G001, S003 S005 · calculationBefore odd cycle edge1 forced next part := sub(1, 1) After odd cycle edge1 forced next part = 0 Conditions: none Compute exactly from original roles and established intermediates,retaining indices,signs,denominators and the operation's carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S004 S006 · impliesBefore cycle traversal index=1;from=1;to=2;normalized actual edge=[1, 2];forced current part=1 After forced next part=0;edge present=Yes Conditions: none Keep traversal order while normalizing each edge.Show both its membership in the complete graph and the forced part flip,including the final return. Step identifiers and dependenciesRule: odd_cycle_every_edge_including_closure · Uses: G001, S005 S007 · calculationBefore odd cycle edge2 forced next part := sub(1, 0) After odd cycle edge2 forced next part = 1 Conditions: none Compute exactly from original roles and established intermediates,retaining indices,signs,denominators and the operation's carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S006 S008 · impliesBefore cycle traversal index=2;from=2;to=0;normalized actual edge=[0, 2];forced current part=0 After forced next part=1;edge present=Yes Conditions: none Keep traversal order while normalizing each edge.Show both its membership in the complete graph and the forced part flip,including the final return. Step identifiers and dependenciesRule: odd_cycle_every_edge_including_closure · Uses: G001, S007 S009 · impliesBefore odd length=3;initial vertex=0 assumed part0;forced closing part=1 After same first vertex would have both parts0 and1;odd_cycle_proves_nonbipartite Conditions: none An odd number of flips returns to the same vertex in the opposite part,a contradiction.No valid bipartition exists,not merely a failed attempt.Extra or isolated vertices do not remove this odd cycle. Step identifiers and dependenciesRule: odd_cycle_parity_contradiction_universal_partition · Uses: G001, S008 ## Reverse checks - 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 — Recompute odd cycle edge2 forced next part with exact rational arithmetic, not rounded decimals. - 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 — Recompute odd cycle edge1 forced next part with exact rational arithmetic, not rounded decimals. - 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 — Recompute odd cycle edge0 forced next part with exact rational arithmetic, not rounded decimals. - 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], [1, 2], [0, 2]], 3, Yes, odd_cycle_proves_nonbipartite) 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: bc873238e80599327c45fd4b20a38441f4965da4713f64e147b5591e61bb4c89 - candidate record SHA-256: 881be9737b89962ce5dbba60d4283bad9894df9116456aba53025324545b5461 - 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-r02-bipartite-graph-0057.json)