Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [0, 2], [3, 4], [3, 5], [1, 4]]. 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], [3, 4], [3, 5], [1, 4]]. 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.
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.
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.
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.
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.