# Complete Bipartitions, Actual Odd Cycles and Isolated-Vertex Boundaries

This batch has passed Schema, deduplication, source-snapshot, and independent-answer recomputation checks; it is still not marked as human-reviewed.

This question bank has corresponding bilingual step-by-step derivations. Source candidate and English ai_candidate states are retained. AI review, finite computation and Lean formal proof are labelled separately.

batch-20261014-advance-20261031-r02-bipartite-graph-foundations · 2026-10-14 · 100 questions

This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.



## 1. Audit This Bipartition without Misclassifying the Entire Graph

Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [2, 3], [3, 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. The total part assignment=[0, 0, 0, 0, 0, 0] uses labels 0 or 1 for every vertex. Its two inverse-image parts are disjoint and cover the whole carrier. Every edge must cross parts; empty parts are allowed in an edgeless graph. Failure of this assignment alone does not prove the graph nonbipartite. Give (parts,conflicting_edges,proper?,classification). List all same-part edges in normalized edge order. Classification is bipartition_certificate or this_partition_fails_not_nonbipartite_proof.

View answer and explanation([0, 0, 0, 0, 0, 0], [[0, 1], [1, 2], [2, 3], [3, 4]], No, this_partition_fails_not_nonbipartite_proof)

Check both endpoints of every complete edge. A passing assignment is a bipartition certificate. A failing assignment may be replaced by a different valid assignment; an odd-cycle witness is a separate obstruction. The exact result, retaining the requested order, is ([0, 0, 0, 0, 0, 0], [[0, 1], [1, 2], [2, 3], [3, 4]], No, this_partition_fails_not_nonbipartite_proof).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r02-bipartite-graph-0001/)

Bipartite graph · qf-mod31-bipartite-partition-audit · candidate





## 2. Audit This Bipartition without Misclassifying the Entire Graph

Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [2, 3], [3, 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. The total part assignment=[1, 0, 0, 0, 0, 0] uses labels 0 or 1 for every vertex. Its two inverse-image parts are disjoint and cover the whole carrier. Every edge must cross parts; empty parts are allowed in an edgeless graph. Failure of this assignment alone does not prove the graph nonbipartite. Give (parts,conflicting_edges,proper?,classification). List all same-part edges in normalized edge order. Classification is bipartition_certificate or this_partition_fails_not_nonbipartite_proof.

View answer and explanation([1, 0, 0, 0, 0, 0], [[1, 2], [2, 3], [3, 4]], No, this_partition_fails_not_nonbipartite_proof)

Check both endpoints of every complete edge. A passing assignment is a bipartition certificate. A failing assignment may be replaced by a different valid assignment; an odd-cycle witness is a separate obstruction. The exact result, retaining the requested order, is ([1, 0, 0, 0, 0, 0], [[1, 2], [2, 3], [3, 4]], No, this_partition_fails_not_nonbipartite_proof).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r02-bipartite-graph-0002/)

Bipartite graph · qf-mod31-bipartite-partition-audit · candidate





## 3. Audit This Bipartition without Misclassifying the Entire Graph

Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [2, 3], [3, 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. The total part assignment=[0, 1, 0, 0, 0, 0] uses labels 0 or 1 for every vertex. Its two inverse-image parts are disjoint and cover the whole carrier. Every edge must cross parts; empty parts are allowed in an edgeless graph. Failure of this assignment alone does not prove the graph nonbipartite. Give (parts,conflicting_edges,proper?,classification). List all same-part edges in normalized edge order. Classification is bipartition_certificate or this_partition_fails_not_nonbipartite_proof.

View answer and explanation([0, 1, 0, 0, 0, 0], [[2, 3], [3, 4]], No, this_partition_fails_not_nonbipartite_proof)

Check both endpoints of every complete edge. A passing assignment is a bipartition certificate. A failing assignment may be replaced by a different valid assignment; an odd-cycle witness is a separate obstruction. The exact result, retaining the requested order, is ([0, 1, 0, 0, 0, 0], [[2, 3], [3, 4]], No, this_partition_fails_not_nonbipartite_proof).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r02-bipartite-graph-0003/)

Bipartite graph · qf-mod31-bipartite-partition-audit · candidate





## 4. Audit This Bipartition without Misclassifying the Entire Graph

Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [2, 3], [3, 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. The total part assignment=[1, 1, 0, 0, 0, 0] uses labels 0 or 1 for every vertex. Its two inverse-image parts are disjoint and cover the whole carrier. Every edge must cross parts; empty parts are allowed in an edgeless graph. Failure of this assignment alone does not prove the graph nonbipartite. Give (parts,conflicting_edges,proper?,classification). List all same-part edges in normalized edge order. Classification is bipartition_certificate or this_partition_fails_not_nonbipartite_proof.

View answer and explanation([1, 1, 0, 0, 0, 0], [[0, 1], [2, 3], [3, 4]], No, this_partition_fails_not_nonbipartite_proof)

Check both endpoints of every complete edge. A passing assignment is a bipartition certificate. A failing assignment may be replaced by a different valid assignment; an odd-cycle witness is a separate obstruction. The exact result, retaining the requested order, is ([1, 1, 0, 0, 0, 0], [[0, 1], [2, 3], [3, 4]], No, this_partition_fails_not_nonbipartite_proof).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r02-bipartite-graph-0004/)

Bipartite graph · qf-mod31-bipartite-partition-audit · candidate





## 5. Audit This Bipartition without Misclassifying the Entire Graph

Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [2, 3], [3, 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. The total part assignment=[0, 0, 1, 0, 0, 0] uses labels 0 or 1 for every vertex. Its two inverse-image parts are disjoint and cover the whole carrier. Every edge must cross parts; empty parts are allowed in an edgeless graph. Failure of this assignment alone does not prove the graph nonbipartite. Give (parts,conflicting_edges,proper?,classification). List all same-part edges in normalized edge order. Classification is bipartition_certificate or this_partition_fails_not_nonbipartite_proof.

View answer and explanation([0, 0, 1, 0, 0, 0], [[0, 1], [3, 4]], No, this_partition_fails_not_nonbipartite_proof)

Check both endpoints of every complete edge. A passing assignment is a bipartition certificate. A failing assignment may be replaced by a different valid assignment; an odd-cycle witness is a separate obstruction. The exact result, retaining the requested order, is ([0, 0, 1, 0, 0, 0], [[0, 1], [3, 4]], No, this_partition_fails_not_nonbipartite_proof).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r02-bipartite-graph-0005/)

Bipartite graph · qf-mod31-bipartite-partition-audit · candidate





## 6. Audit This Bipartition without Misclassifying the Entire Graph

Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [2, 3], [3, 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. The total part assignment=[1, 0, 1, 0, 0, 0] uses labels 0 or 1 for every vertex. Its two inverse-image parts are disjoint and cover the whole carrier. Every edge must cross parts; empty parts are allowed in an edgeless graph. Failure of this assignment alone does not prove the graph nonbipartite. Give (parts,conflicting_edges,proper?,classification). List all same-part edges in normalized edge order. Classification is bipartition_certificate or this_partition_fails_not_nonbipartite_proof.

View answer and explanation([1, 0, 1, 0, 0, 0], [[3, 4]], No, this_partition_fails_not_nonbipartite_proof)

Check both endpoints of every complete edge. A passing assignment is a bipartition certificate. A failing assignment may be replaced by a different valid assignment; an odd-cycle witness is a separate obstruction. The exact result, retaining the requested order, is ([1, 0, 1, 0, 0, 0], [[3, 4]], No, this_partition_fails_not_nonbipartite_proof).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r02-bipartite-graph-0006/)

Bipartite graph · qf-mod31-bipartite-partition-audit · candidate





## 7. Audit This Bipartition without Misclassifying the Entire Graph

Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [2, 3], [3, 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. The total part assignment=[0, 1, 1, 0, 0, 0] uses labels 0 or 1 for every vertex. Its two inverse-image parts are disjoint and cover the whole carrier. Every edge must cross parts; empty parts are allowed in an edgeless graph. Failure of this assignment alone does not prove the graph nonbipartite. Give (parts,conflicting_edges,proper?,classification). List all same-part edges in normalized edge order. Classification is bipartition_certificate or this_partition_fails_not_nonbipartite_proof.

View answer and explanation([0, 1, 1, 0, 0, 0], [[1, 2], [3, 4]], No, this_partition_fails_not_nonbipartite_proof)

Check both endpoints of every complete edge. A passing assignment is a bipartition certificate. A failing assignment may be replaced by a different valid assignment; an odd-cycle witness is a separate obstruction. The exact result, retaining the requested order, is ([0, 1, 1, 0, 0, 0], [[1, 2], [3, 4]], No, this_partition_fails_not_nonbipartite_proof).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r02-bipartite-graph-0007/)

Bipartite graph · qf-mod31-bipartite-partition-audit · candidate





## 8. Audit This Bipartition without Misclassifying the Entire Graph

Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [2, 3], [3, 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. The total part assignment=[1, 1, 1, 0, 0, 0] uses labels 0 or 1 for every vertex. Its two inverse-image parts are disjoint and cover the whole carrier. Every edge must cross parts; empty parts are allowed in an edgeless graph. Failure of this assignment alone does not prove the graph nonbipartite. Give (parts,conflicting_edges,proper?,classification). List all same-part edges in normalized edge order. Classification is bipartition_certificate or this_partition_fails_not_nonbipartite_proof.

View answer and explanation([1, 1, 1, 0, 0, 0], [[0, 1], [1, 2], [3, 4]], No, this_partition_fails_not_nonbipartite_proof)

Check both endpoints of every complete edge. A passing assignment is a bipartition certificate. A failing assignment may be replaced by a different valid assignment; an odd-cycle witness is a separate obstruction. The exact result, retaining the requested order, is ([1, 1, 1, 0, 0, 0], [[0, 1], [1, 2], [3, 4]], No, this_partition_fails_not_nonbipartite_proof).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r02-bipartite-graph-0008/)

Bipartite graph · qf-mod31-bipartite-partition-audit · candidate





## 9. Audit This Bipartition without Misclassifying the Entire Graph

Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [2, 3], [3, 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. The total part assignment=[0, 0, 0, 1, 0, 0] uses labels 0 or 1 for every vertex. Its two inverse-image parts are disjoint and cover the whole carrier. Every edge must cross parts; empty parts are allowed in an edgeless graph. Failure of this assignment alone does not prove the graph nonbipartite. Give (parts,conflicting_edges,proper?,classification). List all same-part edges in normalized edge order. Classification is bipartition_certificate or this_partition_fails_not_nonbipartite_proof.

View answer and explanation([0, 0, 0, 1, 0, 0], [[0, 1], [1, 2]], No, this_partition_fails_not_nonbipartite_proof)

Check both endpoints of every complete edge. A passing assignment is a bipartition certificate. A failing assignment may be replaced by a different valid assignment; an odd-cycle witness is a separate obstruction. The exact result, retaining the requested order, is ([0, 0, 0, 1, 0, 0], [[0, 1], [1, 2]], No, this_partition_fails_not_nonbipartite_proof).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r02-bipartite-graph-0009/)

Bipartite graph · qf-mod31-bipartite-partition-audit · candidate





## 10. Audit This Bipartition without Misclassifying the Entire Graph

Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [2, 3], [3, 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. The total part assignment=[1, 0, 0, 1, 0, 0] uses labels 0 or 1 for every vertex. Its two inverse-image parts are disjoint and cover the whole carrier. Every edge must cross parts; empty parts are allowed in an edgeless graph. Failure of this assignment alone does not prove the graph nonbipartite. Give (parts,conflicting_edges,proper?,classification). List all same-part edges in normalized edge order. Classification is bipartition_certificate or this_partition_fails_not_nonbipartite_proof.

View answer and explanation([1, 0, 0, 1, 0, 0], [[1, 2]], No, this_partition_fails_not_nonbipartite_proof)

Check both endpoints of every complete edge. A passing assignment is a bipartition certificate. A failing assignment may be replaced by a different valid assignment; an odd-cycle witness is a separate obstruction. The exact result, retaining the requested order, is ([1, 0, 0, 1, 0, 0], [[1, 2]], No, this_partition_fails_not_nonbipartite_proof).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r02-bipartite-graph-0010/)

Bipartite graph · qf-mod31-bipartite-partition-audit · candidate





## 11. Audit This Bipartition without Misclassifying the Entire Graph

Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [2, 3], [3, 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. The total part assignment=[0, 1, 0, 1, 0, 0] uses labels 0 or 1 for every vertex. Its two inverse-image parts are disjoint and cover the whole carrier. Every edge must cross parts; empty parts are allowed in an edgeless graph. Failure of this assignment alone does not prove the graph nonbipartite. Give (parts,conflicting_edges,proper?,classification). List all same-part edges in normalized edge order. Classification is bipartition_certificate or this_partition_fails_not_nonbipartite_proof.

View answer and explanation([0, 1, 0, 1, 0, 0], [], Yes, bipartition_certificate)

Check both endpoints of every complete edge. A passing assignment is a bipartition certificate. A failing assignment may be replaced by a different valid assignment; an odd-cycle witness is a separate obstruction. The exact result, retaining the requested order, is ([0, 1, 0, 1, 0, 0], [], Yes, bipartition_certificate).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r02-bipartite-graph-0011/)

Bipartite graph · qf-mod31-bipartite-partition-audit · candidate





## 12. Audit This Bipartition without Misclassifying the Entire Graph

Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [2, 3], [3, 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. The total part assignment=[1, 1, 0, 1, 0, 0] uses labels 0 or 1 for every vertex. Its two inverse-image parts are disjoint and cover the whole carrier. Every edge must cross parts; empty parts are allowed in an edgeless graph. Failure of this assignment alone does not prove the graph nonbipartite. Give (parts,conflicting_edges,proper?,classification). List all same-part edges in normalized edge order. Classification is bipartition_certificate or this_partition_fails_not_nonbipartite_proof.

View answer and explanation([1, 1, 0, 1, 0, 0], [[0, 1]], No, this_partition_fails_not_nonbipartite_proof)

Check both endpoints of every complete edge. A passing assignment is a bipartition certificate. A failing assignment may be replaced by a different valid assignment; an odd-cycle witness is a separate obstruction. The exact result, retaining the requested order, is ([1, 1, 0, 1, 0, 0], [[0, 1]], No, this_partition_fails_not_nonbipartite_proof).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r02-bipartite-graph-0012/)

Bipartite graph · qf-mod31-bipartite-partition-audit · candidate





## 13. Audit This Bipartition without Misclassifying the Entire Graph

Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [2, 3], [3, 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. The total part assignment=[0, 0, 1, 1, 0, 0] uses labels 0 or 1 for every vertex. Its two inverse-image parts are disjoint and cover the whole carrier. Every edge must cross parts; empty parts are allowed in an edgeless graph. Failure of this assignment alone does not prove the graph nonbipartite. Give (parts,conflicting_edges,proper?,classification). List all same-part edges in normalized edge order. Classification is bipartition_certificate or this_partition_fails_not_nonbipartite_proof.

View answer and explanation([0, 0, 1, 1, 0, 0], [[0, 1], [2, 3]], No, this_partition_fails_not_nonbipartite_proof)

Check both endpoints of every complete edge. A passing assignment is a bipartition certificate. A failing assignment may be replaced by a different valid assignment; an odd-cycle witness is a separate obstruction. The exact result, retaining the requested order, is ([0, 0, 1, 1, 0, 0], [[0, 1], [2, 3]], No, this_partition_fails_not_nonbipartite_proof).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r02-bipartite-graph-0013/)

Bipartite graph · qf-mod31-bipartite-partition-audit · candidate





## 14. Audit This Bipartition without Misclassifying the Entire Graph

Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [2, 3], [3, 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. The total part assignment=[1, 0, 1, 1, 0, 0] uses labels 0 or 1 for every vertex. Its two inverse-image parts are disjoint and cover the whole carrier. Every edge must cross parts; empty parts are allowed in an edgeless graph. Failure of this assignment alone does not prove the graph nonbipartite. Give (parts,conflicting_edges,proper?,classification). List all same-part edges in normalized edge order. Classification is bipartition_certificate or this_partition_fails_not_nonbipartite_proof.

View answer and explanation([1, 0, 1, 1, 0, 0], [[2, 3]], No, this_partition_fails_not_nonbipartite_proof)

Check both endpoints of every complete edge. A passing assignment is a bipartition certificate. A failing assignment may be replaced by a different valid assignment; an odd-cycle witness is a separate obstruction. The exact result, retaining the requested order, is ([1, 0, 1, 1, 0, 0], [[2, 3]], No, this_partition_fails_not_nonbipartite_proof).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r02-bipartite-graph-0014/)

Bipartite graph · qf-mod31-bipartite-partition-audit · candidate





## 15. Audit This Bipartition without Misclassifying the Entire Graph

Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [2, 3], [3, 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. The total part assignment=[0, 1, 1, 1, 0, 0] uses labels 0 or 1 for every vertex. Its two inverse-image parts are disjoint and cover the whole carrier. Every edge must cross parts; empty parts are allowed in an edgeless graph. Failure of this assignment alone does not prove the graph nonbipartite. Give (parts,conflicting_edges,proper?,classification). List all same-part edges in normalized edge order. Classification is bipartition_certificate or this_partition_fails_not_nonbipartite_proof.

View answer and explanation([0, 1, 1, 1, 0, 0], [[1, 2], [2, 3]], No, this_partition_fails_not_nonbipartite_proof)

Check both endpoints of every complete edge. A passing assignment is a bipartition certificate. A failing assignment may be replaced by a different valid assignment; an odd-cycle witness is a separate obstruction. The exact result, retaining the requested order, is ([0, 1, 1, 1, 0, 0], [[1, 2], [2, 3]], No, this_partition_fails_not_nonbipartite_proof).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r02-bipartite-graph-0015/)

Bipartite graph · qf-mod31-bipartite-partition-audit · candidate





## 16. Audit This Bipartition without Misclassifying the Entire Graph

Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [2, 3], [3, 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. The total part assignment=[1, 1, 1, 1, 0, 0] uses labels 0 or 1 for every vertex. Its two inverse-image parts are disjoint and cover the whole carrier. Every edge must cross parts; empty parts are allowed in an edgeless graph. Failure of this assignment alone does not prove the graph nonbipartite. Give (parts,conflicting_edges,proper?,classification). List all same-part edges in normalized edge order. Classification is bipartition_certificate or this_partition_fails_not_nonbipartite_proof.

View answer and explanation([1, 1, 1, 1, 0, 0], [[0, 1], [1, 2], [2, 3]], No, this_partition_fails_not_nonbipartite_proof)

Check both endpoints of every complete edge. A passing assignment is a bipartition certificate. A failing assignment may be replaced by a different valid assignment; an odd-cycle witness is a separate obstruction. The exact result, retaining the requested order, is ([1, 1, 1, 1, 0, 0], [[0, 1], [1, 2], [2, 3]], No, this_partition_fails_not_nonbipartite_proof).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r02-bipartite-graph-0016/)

Bipartite graph · qf-mod31-bipartite-partition-audit · candidate





## 17. Audit This Bipartition without Misclassifying the Entire Graph

Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [2, 3], [3, 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. The total part assignment=[0, 0, 0, 0, 1, 0] uses labels 0 or 1 for every vertex. Its two inverse-image parts are disjoint and cover the whole carrier. Every edge must cross parts; empty parts are allowed in an edgeless graph. Failure of this assignment alone does not prove the graph nonbipartite. Give (parts,conflicting_edges,proper?,classification). List all same-part edges in normalized edge order. Classification is bipartition_certificate or this_partition_fails_not_nonbipartite_proof.

View answer and explanation([0, 0, 0, 0, 1, 0], [[0, 1], [1, 2], [2, 3]], No, this_partition_fails_not_nonbipartite_proof)

Check both endpoints of every complete edge. A passing assignment is a bipartition certificate. A failing assignment may be replaced by a different valid assignment; an odd-cycle witness is a separate obstruction. The exact result, retaining the requested order, is ([0, 0, 0, 0, 1, 0], [[0, 1], [1, 2], [2, 3]], No, this_partition_fails_not_nonbipartite_proof).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r02-bipartite-graph-0017/)

Bipartite graph · qf-mod31-bipartite-partition-audit · candidate





## 18. Audit This Bipartition without Misclassifying the Entire Graph

Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [2, 3], [3, 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. The total part assignment=[1, 0, 0, 0, 1, 0] uses labels 0 or 1 for every vertex. Its two inverse-image parts are disjoint and cover the whole carrier. Every edge must cross parts; empty parts are allowed in an edgeless graph. Failure of this assignment alone does not prove the graph nonbipartite. Give (parts,conflicting_edges,proper?,classification). List all same-part edges in normalized edge order. Classification is bipartition_certificate or this_partition_fails_not_nonbipartite_proof.

View answer and explanation([1, 0, 0, 0, 1, 0], [[1, 2], [2, 3]], No, this_partition_fails_not_nonbipartite_proof)

Check both endpoints of every complete edge. A passing assignment is a bipartition certificate. A failing assignment may be replaced by a different valid assignment; an odd-cycle witness is a separate obstruction. The exact result, retaining the requested order, is ([1, 0, 0, 0, 1, 0], [[1, 2], [2, 3]], No, this_partition_fails_not_nonbipartite_proof).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r02-bipartite-graph-0018/)

Bipartite graph · qf-mod31-bipartite-partition-audit · candidate





## 19. Audit This Bipartition without Misclassifying the Entire Graph

Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [2, 3], [3, 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. The total part assignment=[0, 1, 0, 0, 1, 0] uses labels 0 or 1 for every vertex. Its two inverse-image parts are disjoint and cover the whole carrier. Every edge must cross parts; empty parts are allowed in an edgeless graph. Failure of this assignment alone does not prove the graph nonbipartite. Give (parts,conflicting_edges,proper?,classification). List all same-part edges in normalized edge order. Classification is bipartition_certificate or this_partition_fails_not_nonbipartite_proof.

View answer and explanation([0, 1, 0, 0, 1, 0], [[2, 3]], No, this_partition_fails_not_nonbipartite_proof)

Check both endpoints of every complete edge. A passing assignment is a bipartition certificate. A failing assignment may be replaced by a different valid assignment; an odd-cycle witness is a separate obstruction. The exact result, retaining the requested order, is ([0, 1, 0, 0, 1, 0], [[2, 3]], No, this_partition_fails_not_nonbipartite_proof).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r02-bipartite-graph-0019/)

Bipartite graph · qf-mod31-bipartite-partition-audit · candidate





## 20. Audit This Bipartition without Misclassifying the Entire Graph

Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [2, 3], [3, 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. The total part assignment=[1, 1, 0, 0, 1, 0] uses labels 0 or 1 for every vertex. Its two inverse-image parts are disjoint and cover the whole carrier. Every edge must cross parts; empty parts are allowed in an edgeless graph. Failure of this assignment alone does not prove the graph nonbipartite. Give (parts,conflicting_edges,proper?,classification). List all same-part edges in normalized edge order. Classification is bipartition_certificate or this_partition_fails_not_nonbipartite_proof.

View answer and explanation([1, 1, 0, 0, 1, 0], [[0, 1], [2, 3]], No, this_partition_fails_not_nonbipartite_proof)

Check both endpoints of every complete edge. A passing assignment is a bipartition certificate. A failing assignment may be replaced by a different valid assignment; an odd-cycle witness is a separate obstruction. The exact result, retaining the requested order, is ([1, 1, 0, 0, 1, 0], [[0, 1], [2, 3]], No, this_partition_fails_not_nonbipartite_proof).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r02-bipartite-graph-0020/)

Bipartite graph · qf-mod31-bipartite-partition-audit · candidate





## 21. Audit This Bipartition without Misclassifying the Entire Graph

Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [2, 3], [3, 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. The total part assignment=[0, 0, 1, 0, 1, 0] uses labels 0 or 1 for every vertex. Its two inverse-image parts are disjoint and cover the whole carrier. Every edge must cross parts; empty parts are allowed in an edgeless graph. Failure of this assignment alone does not prove the graph nonbipartite. Give (parts,conflicting_edges,proper?,classification). List all same-part edges in normalized edge order. Classification is bipartition_certificate or this_partition_fails_not_nonbipartite_proof.

View answer and explanation([0, 0, 1, 0, 1, 0], [[0, 1]], No, this_partition_fails_not_nonbipartite_proof)

Check both endpoints of every complete edge. A passing assignment is a bipartition certificate. A failing assignment may be replaced by a different valid assignment; an odd-cycle witness is a separate obstruction. The exact result, retaining the requested order, is ([0, 0, 1, 0, 1, 0], [[0, 1]], No, this_partition_fails_not_nonbipartite_proof).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r02-bipartite-graph-0021/)

Bipartite graph · qf-mod31-bipartite-partition-audit · candidate





## 22. Audit This Bipartition without Misclassifying the Entire Graph

Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [2, 3], [3, 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. The total part assignment=[1, 0, 1, 0, 1, 0] uses labels 0 or 1 for every vertex. Its two inverse-image parts are disjoint and cover the whole carrier. Every edge must cross parts; empty parts are allowed in an edgeless graph. Failure of this assignment alone does not prove the graph nonbipartite. Give (parts,conflicting_edges,proper?,classification). List all same-part edges in normalized edge order. Classification is bipartition_certificate or this_partition_fails_not_nonbipartite_proof.

View answer and explanation([1, 0, 1, 0, 1, 0], [], Yes, bipartition_certificate)

Check both endpoints of every complete edge. A passing assignment is a bipartition certificate. A failing assignment may be replaced by a different valid assignment; an odd-cycle witness is a separate obstruction. The exact result, retaining the requested order, is ([1, 0, 1, 0, 1, 0], [], Yes, bipartition_certificate).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r02-bipartite-graph-0022/)

Bipartite graph · qf-mod31-bipartite-partition-audit · candidate





## 23. Audit This Bipartition without Misclassifying the Entire Graph

Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [2, 3], [3, 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. The total part assignment=[0, 1, 1, 0, 1, 0] uses labels 0 or 1 for every vertex. Its two inverse-image parts are disjoint and cover the whole carrier. Every edge must cross parts; empty parts are allowed in an edgeless graph. Failure of this assignment alone does not prove the graph nonbipartite. Give (parts,conflicting_edges,proper?,classification). List all same-part edges in normalized edge order. Classification is bipartition_certificate or this_partition_fails_not_nonbipartite_proof.

View answer and explanation([0, 1, 1, 0, 1, 0], [[1, 2]], No, this_partition_fails_not_nonbipartite_proof)

Check both endpoints of every complete edge. A passing assignment is a bipartition certificate. A failing assignment may be replaced by a different valid assignment; an odd-cycle witness is a separate obstruction. The exact result, retaining the requested order, is ([0, 1, 1, 0, 1, 0], [[1, 2]], No, this_partition_fails_not_nonbipartite_proof).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r02-bipartite-graph-0023/)

Bipartite graph · qf-mod31-bipartite-partition-audit · candidate





## 24. Audit This Bipartition without Misclassifying the Entire Graph

Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [2, 3], [3, 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. The total part assignment=[1, 1, 1, 0, 1, 0] uses labels 0 or 1 for every vertex. Its two inverse-image parts are disjoint and cover the whole carrier. Every edge must cross parts; empty parts are allowed in an edgeless graph. Failure of this assignment alone does not prove the graph nonbipartite. Give (parts,conflicting_edges,proper?,classification). List all same-part edges in normalized edge order. Classification is bipartition_certificate or this_partition_fails_not_nonbipartite_proof.

View answer and explanation([1, 1, 1, 0, 1, 0], [[0, 1], [1, 2]], No, this_partition_fails_not_nonbipartite_proof)

Check both endpoints of every complete edge. A passing assignment is a bipartition certificate. A failing assignment may be replaced by a different valid assignment; an odd-cycle witness is a separate obstruction. The exact result, retaining the requested order, is ([1, 1, 1, 0, 1, 0], [[0, 1], [1, 2]], No, this_partition_fails_not_nonbipartite_proof).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r02-bipartite-graph-0024/)

Bipartite graph · qf-mod31-bipartite-partition-audit · candidate





## 25. Audit This Bipartition without Misclassifying the Entire Graph

Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [2, 3], [3, 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. The total part assignment=[0, 0, 0, 1, 1, 0] uses labels 0 or 1 for every vertex. Its two inverse-image parts are disjoint and cover the whole carrier. Every edge must cross parts; empty parts are allowed in an edgeless graph. Failure of this assignment alone does not prove the graph nonbipartite. Give (parts,conflicting_edges,proper?,classification). List all same-part edges in normalized edge order. Classification is bipartition_certificate or this_partition_fails_not_nonbipartite_proof.

View answer and explanation([0, 0, 0, 1, 1, 0], [[0, 1], [1, 2], [3, 4]], No, this_partition_fails_not_nonbipartite_proof)

Check both endpoints of every complete edge. A passing assignment is a bipartition certificate. A failing assignment may be replaced by a different valid assignment; an odd-cycle witness is a separate obstruction. The exact result, retaining the requested order, is ([0, 0, 0, 1, 1, 0], [[0, 1], [1, 2], [3, 4]], No, this_partition_fails_not_nonbipartite_proof).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r02-bipartite-graph-0025/)

Bipartite graph · qf-mod31-bipartite-partition-audit · candidate





## 26. One Edge Crossing Is Separate from the Whole Partition Audit

Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [2, 3], [3, 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. The total part assignment=[0, 0, 0, 0, 0, 0] uses labels 0 or 1 for every vertex. Its two inverse-image parts are disjoint and cover the whole carrier. Every edge must cross parts; empty parts are allowed in an edgeless graph. Failure of this assignment alone does not prove the graph nonbipartite. edge_index=0 selects the normalized sorted edge list, not the raw input order. Give (edge_index,edge,left_part,right_part,crosses?,whole_partition_proper?).

View answer and explanation(0, [0, 1], 0, 0, No, No)

Read the selected normalized edge and report its two part labels, then retain the independent all-edge properness flag. One crossing edge cannot certify all remaining edges. The exact result, retaining the requested order, is (0, [0, 1], 0, 0, No, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r02-bipartite-graph-0026/)

Bipartite graph · qf-mod31-bipartite-edge-crossing · candidate





## 27. One Edge Crossing Is Separate from the Whole Partition Audit

Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [2, 3], [3, 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. The total part assignment=[1, 0, 0, 0, 0, 0] uses labels 0 or 1 for every vertex. Its two inverse-image parts are disjoint and cover the whole carrier. Every edge must cross parts; empty parts are allowed in an edgeless graph. Failure of this assignment alone does not prove the graph nonbipartite. edge_index=1 selects the normalized sorted edge list, not the raw input order. Give (edge_index,edge,left_part,right_part,crosses?,whole_partition_proper?).

View answer and explanation(1, [1, 2], 0, 0, No, No)

Read the selected normalized edge and report its two part labels, then retain the independent all-edge properness flag. One crossing edge cannot certify all remaining edges. The exact result, retaining the requested order, is (1, [1, 2], 0, 0, No, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r02-bipartite-graph-0027/)

Bipartite graph · qf-mod31-bipartite-edge-crossing · candidate





## 28. One Edge Crossing Is Separate from the Whole Partition Audit

Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [2, 3], [3, 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. The total part assignment=[0, 1, 0, 0, 0, 0] uses labels 0 or 1 for every vertex. Its two inverse-image parts are disjoint and cover the whole carrier. Every edge must cross parts; empty parts are allowed in an edgeless graph. Failure of this assignment alone does not prove the graph nonbipartite. edge_index=2 selects the normalized sorted edge list, not the raw input order. Give (edge_index,edge,left_part,right_part,crosses?,whole_partition_proper?).

View answer and explanation(2, [2, 3], 0, 0, No, No)

Read the selected normalized edge and report its two part labels, then retain the independent all-edge properness flag. One crossing edge cannot certify all remaining edges. The exact result, retaining the requested order, is (2, [2, 3], 0, 0, No, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r02-bipartite-graph-0028/)

Bipartite graph · qf-mod31-bipartite-edge-crossing · candidate





## 29. One Edge Crossing Is Separate from the Whole Partition Audit

Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [2, 3], [3, 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. The total part assignment=[1, 1, 0, 0, 0, 0] uses labels 0 or 1 for every vertex. Its two inverse-image parts are disjoint and cover the whole carrier. Every edge must cross parts; empty parts are allowed in an edgeless graph. Failure of this assignment alone does not prove the graph nonbipartite. edge_index=3 selects the normalized sorted edge list, not the raw input order. Give (edge_index,edge,left_part,right_part,crosses?,whole_partition_proper?).

View answer and explanation(3, [3, 4], 0, 0, No, No)

Read the selected normalized edge and report its two part labels, then retain the independent all-edge properness flag. One crossing edge cannot certify all remaining edges. The exact result, retaining the requested order, is (3, [3, 4], 0, 0, No, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r02-bipartite-graph-0029/)

Bipartite graph · qf-mod31-bipartite-edge-crossing · candidate





## 30. One Edge Crossing Is Separate from the Whole Partition Audit

Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [2, 3], [3, 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. The total part assignment=[0, 0, 1, 0, 0, 0] uses labels 0 or 1 for every vertex. Its two inverse-image parts are disjoint and cover the whole carrier. Every edge must cross parts; empty parts are allowed in an edgeless graph. Failure of this assignment alone does not prove the graph nonbipartite. edge_index=0 selects the normalized sorted edge list, not the raw input order. Give (edge_index,edge,left_part,right_part,crosses?,whole_partition_proper?).

View answer and explanation(0, [0, 1], 0, 0, No, No)

Read the selected normalized edge and report its two part labels, then retain the independent all-edge properness flag. One crossing edge cannot certify all remaining edges. The exact result, retaining the requested order, is (0, [0, 1], 0, 0, No, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r02-bipartite-graph-0030/)

Bipartite graph · qf-mod31-bipartite-edge-crossing · candidate





## 31. One Edge Crossing Is Separate from the Whole Partition Audit

Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [2, 3], [3, 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. The total part assignment=[1, 0, 1, 0, 0, 0] uses labels 0 or 1 for every vertex. Its two inverse-image parts are disjoint and cover the whole carrier. Every edge must cross parts; empty parts are allowed in an edgeless graph. Failure of this assignment alone does not prove the graph nonbipartite. edge_index=1 selects the normalized sorted edge list, not the raw input order. Give (edge_index,edge,left_part,right_part,crosses?,whole_partition_proper?).

View answer and explanation(1, [1, 2], 0, 1, Yes, No)

Read the selected normalized edge and report its two part labels, then retain the independent all-edge properness flag. One crossing edge cannot certify all remaining edges. The exact result, retaining the requested order, is (1, [1, 2], 0, 1, Yes, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r02-bipartite-graph-0031/)

Bipartite graph · qf-mod31-bipartite-edge-crossing · candidate





## 32. One Edge Crossing Is Separate from the Whole Partition Audit

Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [2, 3], [3, 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. The total part assignment=[0, 1, 1, 0, 0, 0] uses labels 0 or 1 for every vertex. Its two inverse-image parts are disjoint and cover the whole carrier. Every edge must cross parts; empty parts are allowed in an edgeless graph. Failure of this assignment alone does not prove the graph nonbipartite. edge_index=2 selects the normalized sorted edge list, not the raw input order. Give (edge_index,edge,left_part,right_part,crosses?,whole_partition_proper?).

View answer and explanation(2, [2, 3], 1, 0, Yes, No)

Read the selected normalized edge and report its two part labels, then retain the independent all-edge properness flag. One crossing edge cannot certify all remaining edges. The exact result, retaining the requested order, is (2, [2, 3], 1, 0, Yes, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r02-bipartite-graph-0032/)

Bipartite graph · qf-mod31-bipartite-edge-crossing · candidate





## 33. One Edge Crossing Is Separate from the Whole Partition Audit

Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [2, 3], [3, 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. The total part assignment=[1, 1, 1, 0, 0, 0] uses labels 0 or 1 for every vertex. Its two inverse-image parts are disjoint and cover the whole carrier. Every edge must cross parts; empty parts are allowed in an edgeless graph. Failure of this assignment alone does not prove the graph nonbipartite. edge_index=3 selects the normalized sorted edge list, not the raw input order. Give (edge_index,edge,left_part,right_part,crosses?,whole_partition_proper?).

View answer and explanation(3, [3, 4], 0, 0, No, No)

Read the selected normalized edge and report its two part labels, then retain the independent all-edge properness flag. One crossing edge cannot certify all remaining edges. The exact result, retaining the requested order, is (3, [3, 4], 0, 0, No, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r02-bipartite-graph-0033/)

Bipartite graph · qf-mod31-bipartite-edge-crossing · candidate





## 34. One Edge Crossing Is Separate from the Whole Partition Audit

Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [2, 3], [3, 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. The total part assignment=[0, 0, 0, 1, 0, 0] uses labels 0 or 1 for every vertex. Its two inverse-image parts are disjoint and cover the whole carrier. Every edge must cross parts; empty parts are allowed in an edgeless graph. Failure of this assignment alone does not prove the graph nonbipartite. edge_index=0 selects the normalized sorted edge list, not the raw input order. Give (edge_index,edge,left_part,right_part,crosses?,whole_partition_proper?).

View answer and explanation(0, [0, 1], 0, 0, No, No)

Read the selected normalized edge and report its two part labels, then retain the independent all-edge properness flag. One crossing edge cannot certify all remaining edges. The exact result, retaining the requested order, is (0, [0, 1], 0, 0, No, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r02-bipartite-graph-0034/)

Bipartite graph · qf-mod31-bipartite-edge-crossing · candidate





## 35. One Edge Crossing Is Separate from the Whole Partition Audit

Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [2, 3], [3, 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. The total part assignment=[1, 0, 0, 1, 0, 0] uses labels 0 or 1 for every vertex. Its two inverse-image parts are disjoint and cover the whole carrier. Every edge must cross parts; empty parts are allowed in an edgeless graph. Failure of this assignment alone does not prove the graph nonbipartite. edge_index=1 selects the normalized sorted edge list, not the raw input order. Give (edge_index,edge,left_part,right_part,crosses?,whole_partition_proper?).

View answer and explanation(1, [1, 2], 0, 0, No, No)

Read the selected normalized edge and report its two part labels, then retain the independent all-edge properness flag. One crossing edge cannot certify all remaining edges. The exact result, retaining the requested order, is (1, [1, 2], 0, 0, No, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r02-bipartite-graph-0035/)

Bipartite graph · qf-mod31-bipartite-edge-crossing · candidate





## 36. One Edge Crossing Is Separate from the Whole Partition Audit

Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [2, 3], [3, 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. The total part assignment=[0, 1, 0, 1, 0, 0] uses labels 0 or 1 for every vertex. Its two inverse-image parts are disjoint and cover the whole carrier. Every edge must cross parts; empty parts are allowed in an edgeless graph. Failure of this assignment alone does not prove the graph nonbipartite. edge_index=2 selects the normalized sorted edge list, not the raw input order. Give (edge_index,edge,left_part,right_part,crosses?,whole_partition_proper?).

View answer and explanation(2, [2, 3], 0, 1, Yes, Yes)

Read the selected normalized edge and report its two part labels, then retain the independent all-edge properness flag. One crossing edge cannot certify all remaining edges. The exact result, retaining the requested order, is (2, [2, 3], 0, 1, Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r02-bipartite-graph-0036/)

Bipartite graph · qf-mod31-bipartite-edge-crossing · candidate





## 37. One Edge Crossing Is Separate from the Whole Partition Audit

Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [2, 3], [3, 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. The total part assignment=[1, 1, 0, 1, 0, 0] uses labels 0 or 1 for every vertex. Its two inverse-image parts are disjoint and cover the whole carrier. Every edge must cross parts; empty parts are allowed in an edgeless graph. Failure of this assignment alone does not prove the graph nonbipartite. edge_index=3 selects the normalized sorted edge list, not the raw input order. Give (edge_index,edge,left_part,right_part,crosses?,whole_partition_proper?).

View answer and explanation(3, [3, 4], 1, 0, Yes, No)

Read the selected normalized edge and report its two part labels, then retain the independent all-edge properness flag. One crossing edge cannot certify all remaining edges. The exact result, retaining the requested order, is (3, [3, 4], 1, 0, Yes, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r02-bipartite-graph-0037/)

Bipartite graph · qf-mod31-bipartite-edge-crossing · candidate





## 38. One Edge Crossing Is Separate from the Whole Partition Audit

Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [2, 3], [3, 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. The total part assignment=[0, 0, 1, 1, 0, 0] uses labels 0 or 1 for every vertex. Its two inverse-image parts are disjoint and cover the whole carrier. Every edge must cross parts; empty parts are allowed in an edgeless graph. Failure of this assignment alone does not prove the graph nonbipartite. edge_index=0 selects the normalized sorted edge list, not the raw input order. Give (edge_index,edge,left_part,right_part,crosses?,whole_partition_proper?).

View answer and explanation(0, [0, 1], 0, 0, No, No)

Read the selected normalized edge and report its two part labels, then retain the independent all-edge properness flag. One crossing edge cannot certify all remaining edges. The exact result, retaining the requested order, is (0, [0, 1], 0, 0, No, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r02-bipartite-graph-0038/)

Bipartite graph · qf-mod31-bipartite-edge-crossing · candidate





## 39. One Edge Crossing Is Separate from the Whole Partition Audit

Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [2, 3], [3, 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. The total part assignment=[1, 0, 1, 1, 0, 0] uses labels 0 or 1 for every vertex. Its two inverse-image parts are disjoint and cover the whole carrier. Every edge must cross parts; empty parts are allowed in an edgeless graph. Failure of this assignment alone does not prove the graph nonbipartite. edge_index=1 selects the normalized sorted edge list, not the raw input order. Give (edge_index,edge,left_part,right_part,crosses?,whole_partition_proper?).

View answer and explanation(1, [1, 2], 0, 1, Yes, No)

Read the selected normalized edge and report its two part labels, then retain the independent all-edge properness flag. One crossing edge cannot certify all remaining edges. The exact result, retaining the requested order, is (1, [1, 2], 0, 1, Yes, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r02-bipartite-graph-0039/)

Bipartite graph · qf-mod31-bipartite-edge-crossing · candidate





## 40. One Edge Crossing Is Separate from the Whole Partition Audit

Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [2, 3], [3, 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. The total part assignment=[0, 1, 1, 1, 0, 0] uses labels 0 or 1 for every vertex. Its two inverse-image parts are disjoint and cover the whole carrier. Every edge must cross parts; empty parts are allowed in an edgeless graph. Failure of this assignment alone does not prove the graph nonbipartite. edge_index=2 selects the normalized sorted edge list, not the raw input order. Give (edge_index,edge,left_part,right_part,crosses?,whole_partition_proper?).

View answer and explanation(2, [2, 3], 1, 1, No, No)

Read the selected normalized edge and report its two part labels, then retain the independent all-edge properness flag. One crossing edge cannot certify all remaining edges. The exact result, retaining the requested order, is (2, [2, 3], 1, 1, No, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r02-bipartite-graph-0040/)

Bipartite graph · qf-mod31-bipartite-edge-crossing · candidate





## 41. One Edge Crossing Is Separate from the Whole Partition Audit

Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [2, 3], [3, 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. The total part assignment=[1, 1, 1, 1, 0, 0] uses labels 0 or 1 for every vertex. Its two inverse-image parts are disjoint and cover the whole carrier. Every edge must cross parts; empty parts are allowed in an edgeless graph. Failure of this assignment alone does not prove the graph nonbipartite. edge_index=3 selects the normalized sorted edge list, not the raw input order. Give (edge_index,edge,left_part,right_part,crosses?,whole_partition_proper?).

View answer and explanation(3, [3, 4], 1, 0, Yes, No)

Read the selected normalized edge and report its two part labels, then retain the independent all-edge properness flag. One crossing edge cannot certify all remaining edges. The exact result, retaining the requested order, is (3, [3, 4], 1, 0, Yes, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r02-bipartite-graph-0041/)

Bipartite graph · qf-mod31-bipartite-edge-crossing · candidate





## 42. One Edge Crossing Is Separate from the Whole Partition Audit

Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [2, 3], [3, 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. The total part assignment=[0, 0, 0, 0, 1, 0] uses labels 0 or 1 for every vertex. Its two inverse-image parts are disjoint and cover the whole carrier. Every edge must cross parts; empty parts are allowed in an edgeless graph. Failure of this assignment alone does not prove the graph nonbipartite. edge_index=0 selects the normalized sorted edge list, not the raw input order. Give (edge_index,edge,left_part,right_part,crosses?,whole_partition_proper?).

View answer and explanation(0, [0, 1], 0, 0, No, No)

Read the selected normalized edge and report its two part labels, then retain the independent all-edge properness flag. One crossing edge cannot certify all remaining edges. The exact result, retaining the requested order, is (0, [0, 1], 0, 0, No, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r02-bipartite-graph-0042/)

Bipartite graph · qf-mod31-bipartite-edge-crossing · candidate





## 43. One Edge Crossing Is Separate from the Whole Partition Audit

Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [2, 3], [3, 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. The total part assignment=[1, 0, 0, 0, 1, 0] uses labels 0 or 1 for every vertex. Its two inverse-image parts are disjoint and cover the whole carrier. Every edge must cross parts; empty parts are allowed in an edgeless graph. Failure of this assignment alone does not prove the graph nonbipartite. edge_index=1 selects the normalized sorted edge list, not the raw input order. Give (edge_index,edge,left_part,right_part,crosses?,whole_partition_proper?).

View answer and explanation(1, [1, 2], 0, 0, No, No)

Read the selected normalized edge and report its two part labels, then retain the independent all-edge properness flag. One crossing edge cannot certify all remaining edges. The exact result, retaining the requested order, is (1, [1, 2], 0, 0, No, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r02-bipartite-graph-0043/)

Bipartite graph · qf-mod31-bipartite-edge-crossing · candidate





## 44. One Edge Crossing Is Separate from the Whole Partition Audit

Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [2, 3], [3, 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. The total part assignment=[0, 1, 0, 0, 1, 0] uses labels 0 or 1 for every vertex. Its two inverse-image parts are disjoint and cover the whole carrier. Every edge must cross parts; empty parts are allowed in an edgeless graph. Failure of this assignment alone does not prove the graph nonbipartite. edge_index=2 selects the normalized sorted edge list, not the raw input order. Give (edge_index,edge,left_part,right_part,crosses?,whole_partition_proper?).

View answer and explanation(2, [2, 3], 0, 0, No, No)

Read the selected normalized edge and report its two part labels, then retain the independent all-edge properness flag. One crossing edge cannot certify all remaining edges. The exact result, retaining the requested order, is (2, [2, 3], 0, 0, No, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r02-bipartite-graph-0044/)

Bipartite graph · qf-mod31-bipartite-edge-crossing · candidate





## 45. One Edge Crossing Is Separate from the Whole Partition Audit

Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [2, 3], [3, 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. The total part assignment=[1, 1, 0, 0, 1, 0] uses labels 0 or 1 for every vertex. Its two inverse-image parts are disjoint and cover the whole carrier. Every edge must cross parts; empty parts are allowed in an edgeless graph. Failure of this assignment alone does not prove the graph nonbipartite. edge_index=3 selects the normalized sorted edge list, not the raw input order. Give (edge_index,edge,left_part,right_part,crosses?,whole_partition_proper?).

View answer and explanation(3, [3, 4], 0, 1, Yes, No)

Read the selected normalized edge and report its two part labels, then retain the independent all-edge properness flag. One crossing edge cannot certify all remaining edges. The exact result, retaining the requested order, is (3, [3, 4], 0, 1, Yes, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r02-bipartite-graph-0045/)

Bipartite graph · qf-mod31-bipartite-edge-crossing · candidate





## 46. One Edge Crossing Is Separate from the Whole Partition Audit

Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [2, 3], [3, 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. The total part assignment=[0, 0, 1, 0, 1, 0] uses labels 0 or 1 for every vertex. Its two inverse-image parts are disjoint and cover the whole carrier. Every edge must cross parts; empty parts are allowed in an edgeless graph. Failure of this assignment alone does not prove the graph nonbipartite. edge_index=0 selects the normalized sorted edge list, not the raw input order. Give (edge_index,edge,left_part,right_part,crosses?,whole_partition_proper?).

View answer and explanation(0, [0, 1], 0, 0, No, No)

Read the selected normalized edge and report its two part labels, then retain the independent all-edge properness flag. One crossing edge cannot certify all remaining edges. The exact result, retaining the requested order, is (0, [0, 1], 0, 0, No, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r02-bipartite-graph-0046/)

Bipartite graph · qf-mod31-bipartite-edge-crossing · candidate





## 47. One Edge Crossing Is Separate from the Whole Partition Audit

Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [2, 3], [3, 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. The total part assignment=[1, 0, 1, 0, 1, 0] uses labels 0 or 1 for every vertex. Its two inverse-image parts are disjoint and cover the whole carrier. Every edge must cross parts; empty parts are allowed in an edgeless graph. Failure of this assignment alone does not prove the graph nonbipartite. edge_index=1 selects the normalized sorted edge list, not the raw input order. Give (edge_index,edge,left_part,right_part,crosses?,whole_partition_proper?).

View answer and explanation(1, [1, 2], 0, 1, Yes, Yes)

Read the selected normalized edge and report its two part labels, then retain the independent all-edge properness flag. One crossing edge cannot certify all remaining edges. The exact result, retaining the requested order, is (1, [1, 2], 0, 1, Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r02-bipartite-graph-0047/)

Bipartite graph · qf-mod31-bipartite-edge-crossing · candidate





## 48. One Edge Crossing Is Separate from the Whole Partition Audit

Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [2, 3], [3, 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. The total part assignment=[0, 1, 1, 0, 1, 0] uses labels 0 or 1 for every vertex. Its two inverse-image parts are disjoint and cover the whole carrier. Every edge must cross parts; empty parts are allowed in an edgeless graph. Failure of this assignment alone does not prove the graph nonbipartite. edge_index=2 selects the normalized sorted edge list, not the raw input order. Give (edge_index,edge,left_part,right_part,crosses?,whole_partition_proper?).

View answer and explanation(2, [2, 3], 1, 0, Yes, No)

Read the selected normalized edge and report its two part labels, then retain the independent all-edge properness flag. One crossing edge cannot certify all remaining edges. The exact result, retaining the requested order, is (2, [2, 3], 1, 0, Yes, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r02-bipartite-graph-0048/)

Bipartite graph · qf-mod31-bipartite-edge-crossing · candidate





## 49. One Edge Crossing Is Separate from the Whole Partition Audit

Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [2, 3], [3, 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. The total part assignment=[1, 1, 1, 0, 1, 0] uses labels 0 or 1 for every vertex. Its two inverse-image parts are disjoint and cover the whole carrier. Every edge must cross parts; empty parts are allowed in an edgeless graph. Failure of this assignment alone does not prove the graph nonbipartite. edge_index=3 selects the normalized sorted edge list, not the raw input order. Give (edge_index,edge,left_part,right_part,crosses?,whole_partition_proper?).

View answer and explanation(3, [3, 4], 0, 1, Yes, No)

Read the selected normalized edge and report its two part labels, then retain the independent all-edge properness flag. One crossing edge cannot certify all remaining edges. The exact result, retaining the requested order, is (3, [3, 4], 0, 1, Yes, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r02-bipartite-graph-0049/)

Bipartite graph · qf-mod31-bipartite-edge-crossing · candidate





## 50. One Edge Crossing Is Separate from the Whole Partition Audit

Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [2, 3], [3, 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. The total part assignment=[0, 0, 0, 1, 1, 0] uses labels 0 or 1 for every vertex. Its two inverse-image parts are disjoint and cover the whole carrier. Every edge must cross parts; empty parts are allowed in an edgeless graph. Failure of this assignment alone does not prove the graph nonbipartite. edge_index=0 selects the normalized sorted edge list, not the raw input order. Give (edge_index,edge,left_part,right_part,crosses?,whole_partition_proper?).

View answer and explanation(0, [0, 1], 0, 0, No, No)

Read the selected normalized edge and report its two part labels, then retain the independent all-edge properness flag. One crossing edge cannot certify all remaining edges. The exact result, retaining the requested order, is (0, [0, 1], 0, 0, No, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r02-bipartite-graph-0050/)

Bipartite graph · qf-mod31-bipartite-edge-crossing · candidate





## 51. An Actual Simple Odd Cycle Certifies Nonbipartiteness

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

View answer and explanation([0, 1, 2], [[0, 1], [1, 2], [0, 2]], 3, Yes, odd_cycle_proves_nonbipartite)

Alternating between two parts around an odd cycle cannot return consistently to the initial part. The certificate requires the full actual cycle, not merely a failed candidate partition or an incomplete walk. The exact result, retaining the requested order, is ([0, 1, 2], [[0, 1], [1, 2], [0, 2]], 3, Yes, odd_cycle_proves_nonbipartite).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r02-bipartite-graph-0051/)

Bipartite graph · qf-mod31-bipartite-odd-cycle · candidate





## 52. An Actual Simple Odd Cycle Certifies Nonbipartiteness

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

View answer and explanation([0, 1, 2], [[0, 1], [1, 2], [0, 2]], 3, Yes, odd_cycle_proves_nonbipartite)

Alternating between two parts around an odd cycle cannot return consistently to the initial part. The certificate requires the full actual cycle, not merely a failed candidate partition or an incomplete walk. The exact result, retaining the requested order, is ([0, 1, 2], [[0, 1], [1, 2], [0, 2]], 3, Yes, odd_cycle_proves_nonbipartite).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r02-bipartite-graph-0052/)

Bipartite graph · qf-mod31-bipartite-odd-cycle · candidate





## 53. An Actual Simple Odd Cycle Certifies Nonbipartiteness

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

View answer and explanation([0, 1, 2], [[0, 1], [1, 2], [0, 2]], 3, Yes, odd_cycle_proves_nonbipartite)

Alternating between two parts around an odd cycle cannot return consistently to the initial part. The certificate requires the full actual cycle, not merely a failed candidate partition or an incomplete walk. The exact result, retaining the requested order, is ([0, 1, 2], [[0, 1], [1, 2], [0, 2]], 3, Yes, odd_cycle_proves_nonbipartite).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r02-bipartite-graph-0053/)

Bipartite graph · qf-mod31-bipartite-odd-cycle · candidate





## 54. An Actual Simple Odd Cycle Certifies Nonbipartiteness

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

View answer and explanation([0, 1, 2], [[0, 1], [1, 2], [0, 2]], 3, Yes, odd_cycle_proves_nonbipartite)

Alternating between two parts around an odd cycle cannot return consistently to the initial part. The certificate requires the full actual cycle, not merely a failed candidate partition or an incomplete walk. The exact result, retaining the requested order, is ([0, 1, 2], [[0, 1], [1, 2], [0, 2]], 3, Yes, odd_cycle_proves_nonbipartite).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r02-bipartite-graph-0054/)

Bipartite graph · qf-mod31-bipartite-odd-cycle · candidate





## 55. An Actual Simple Odd Cycle Certifies Nonbipartiteness

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

View answer and explanation([0, 1, 2], [[0, 1], [1, 2], [0, 2]], 3, Yes, odd_cycle_proves_nonbipartite)

Alternating between two parts around an odd cycle cannot return consistently to the initial part. The certificate requires the full actual cycle, not merely a failed candidate partition or an incomplete walk. The exact result, retaining the requested order, is ([0, 1, 2], [[0, 1], [1, 2], [0, 2]], 3, Yes, odd_cycle_proves_nonbipartite).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r02-bipartite-graph-0055/)

Bipartite graph · qf-mod31-bipartite-odd-cycle · candidate





## 56. An Actual Simple Odd Cycle Certifies Nonbipartiteness

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]]. 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.

View answer and explanation([0, 1, 2], [[0, 1], [1, 2], [0, 2]], 3, Yes, odd_cycle_proves_nonbipartite)

Alternating between two parts around an odd cycle cannot return consistently to the initial part. The certificate requires the full actual cycle, not merely a failed candidate partition or an incomplete walk. The exact result, retaining the requested order, is ([0, 1, 2], [[0, 1], [1, 2], [0, 2]], 3, Yes, odd_cycle_proves_nonbipartite).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r02-bipartite-graph-0056/)

Bipartite graph · qf-mod31-bipartite-odd-cycle · candidate





## 57. An Actual Simple Odd Cycle Certifies Nonbipartiteness

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.

View answer and explanation([0, 1, 2], [[0, 1], [1, 2], [0, 2]], 3, Yes, odd_cycle_proves_nonbipartite)

Alternating between two parts around an odd cycle cannot return consistently to the initial part. The certificate requires the full actual cycle, not merely a failed candidate partition or an incomplete walk. The exact result, retaining the requested order, is ([0, 1, 2], [[0, 1], [1, 2], [0, 2]], 3, Yes, odd_cycle_proves_nonbipartite).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r02-bipartite-graph-0057/)

Bipartite graph · qf-mod31-bipartite-odd-cycle · candidate





## 58. An Actual Simple Odd Cycle Certifies Nonbipartiteness

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

View answer and explanation([0, 1, 2], [[0, 1], [1, 2], [0, 2]], 3, Yes, odd_cycle_proves_nonbipartite)

Alternating between two parts around an odd cycle cannot return consistently to the initial part. The certificate requires the full actual cycle, not merely a failed candidate partition or an incomplete walk. The exact result, retaining the requested order, is ([0, 1, 2], [[0, 1], [1, 2], [0, 2]], 3, Yes, odd_cycle_proves_nonbipartite).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r02-bipartite-graph-0058/)

Bipartite graph · qf-mod31-bipartite-odd-cycle · candidate





## 59. An Actual Simple Odd Cycle Certifies Nonbipartiteness

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

View answer and explanation([0, 1, 2], [[0, 1], [1, 2], [0, 2]], 3, Yes, odd_cycle_proves_nonbipartite)

Alternating between two parts around an odd cycle cannot return consistently to the initial part. The certificate requires the full actual cycle, not merely a failed candidate partition or an incomplete walk. The exact result, retaining the requested order, is ([0, 1, 2], [[0, 1], [1, 2], [0, 2]], 3, Yes, odd_cycle_proves_nonbipartite).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r02-bipartite-graph-0059/)

Bipartite graph · qf-mod31-bipartite-odd-cycle · candidate





## 60. An Actual Simple Odd Cycle Certifies Nonbipartiteness

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

View answer and explanation([0, 1, 2], [[0, 1], [1, 2], [0, 2]], 3, Yes, odd_cycle_proves_nonbipartite)

Alternating between two parts around an odd cycle cannot return consistently to the initial part. The certificate requires the full actual cycle, not merely a failed candidate partition or an incomplete walk. The exact result, retaining the requested order, is ([0, 1, 2], [[0, 1], [1, 2], [0, 2]], 3, Yes, odd_cycle_proves_nonbipartite).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r02-bipartite-graph-0060/)

Bipartite graph · qf-mod31-bipartite-odd-cycle · candidate





## 61. An Actual Simple Odd Cycle Certifies Nonbipartiteness

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

View answer and explanation([0, 1, 2], [[0, 1], [1, 2], [0, 2]], 3, Yes, odd_cycle_proves_nonbipartite)

Alternating between two parts around an odd cycle cannot return consistently to the initial part. The certificate requires the full actual cycle, not merely a failed candidate partition or an incomplete walk. The exact result, retaining the requested order, is ([0, 1, 2], [[0, 1], [1, 2], [0, 2]], 3, Yes, odd_cycle_proves_nonbipartite).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r02-bipartite-graph-0061/)

Bipartite graph · qf-mod31-bipartite-odd-cycle · candidate





## 62. An Actual Simple Odd Cycle Certifies Nonbipartiteness

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

View answer and explanation([0, 1, 2], [[0, 1], [1, 2], [0, 2]], 3, Yes, odd_cycle_proves_nonbipartite)

Alternating between two parts around an odd cycle cannot return consistently to the initial part. The certificate requires the full actual cycle, not merely a failed candidate partition or an incomplete walk. The exact result, retaining the requested order, is ([0, 1, 2], [[0, 1], [1, 2], [0, 2]], 3, Yes, odd_cycle_proves_nonbipartite).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r02-bipartite-graph-0062/)

Bipartite graph · qf-mod31-bipartite-odd-cycle · candidate





## 63. An Actual Simple Odd Cycle Certifies Nonbipartiteness

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

View answer and explanation([0, 1, 2], [[0, 1], [1, 2], [0, 2]], 3, Yes, odd_cycle_proves_nonbipartite)

Alternating between two parts around an odd cycle cannot return consistently to the initial part. The certificate requires the full actual cycle, not merely a failed candidate partition or an incomplete walk. The exact result, retaining the requested order, is ([0, 1, 2], [[0, 1], [1, 2], [0, 2]], 3, Yes, odd_cycle_proves_nonbipartite).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r02-bipartite-graph-0063/)

Bipartite graph · qf-mod31-bipartite-odd-cycle · candidate





## 64. An Actual Simple Odd Cycle Certifies Nonbipartiteness

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], [0, 3]]. 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.

View answer and explanation([0, 1, 2], [[0, 1], [1, 2], [0, 2]], 3, Yes, odd_cycle_proves_nonbipartite)

Alternating between two parts around an odd cycle cannot return consistently to the initial part. The certificate requires the full actual cycle, not merely a failed candidate partition or an incomplete walk. The exact result, retaining the requested order, is ([0, 1, 2], [[0, 1], [1, 2], [0, 2]], 3, Yes, odd_cycle_proves_nonbipartite).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r02-bipartite-graph-0064/)

Bipartite graph · qf-mod31-bipartite-odd-cycle · candidate





## 65. An Actual Simple Odd Cycle Certifies Nonbipartiteness

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], [0, 3]]. 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.

View answer and explanation([0, 1, 2], [[0, 1], [1, 2], [0, 2]], 3, Yes, odd_cycle_proves_nonbipartite)

Alternating between two parts around an odd cycle cannot return consistently to the initial part. The certificate requires the full actual cycle, not merely a failed candidate partition or an incomplete walk. The exact result, retaining the requested order, is ([0, 1, 2], [[0, 1], [1, 2], [0, 2]], 3, Yes, odd_cycle_proves_nonbipartite).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r02-bipartite-graph-0065/)

Bipartite graph · qf-mod31-bipartite-odd-cycle · candidate





## 66. An Actual Simple Odd Cycle Certifies Nonbipartiteness

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

View answer and explanation([0, 1, 2], [[0, 1], [1, 2], [0, 2]], 3, Yes, odd_cycle_proves_nonbipartite)

Alternating between two parts around an odd cycle cannot return consistently to the initial part. The certificate requires the full actual cycle, not merely a failed candidate partition or an incomplete walk. The exact result, retaining the requested order, is ([0, 1, 2], [[0, 1], [1, 2], [0, 2]], 3, Yes, odd_cycle_proves_nonbipartite).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r02-bipartite-graph-0066/)

Bipartite graph · qf-mod31-bipartite-odd-cycle · candidate





## 67. An Actual Simple Odd Cycle Certifies Nonbipartiteness

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

View answer and explanation([0, 1, 2], [[0, 1], [1, 2], [0, 2]], 3, Yes, odd_cycle_proves_nonbipartite)

Alternating between two parts around an odd cycle cannot return consistently to the initial part. The certificate requires the full actual cycle, not merely a failed candidate partition or an incomplete walk. The exact result, retaining the requested order, is ([0, 1, 2], [[0, 1], [1, 2], [0, 2]], 3, Yes, odd_cycle_proves_nonbipartite).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r02-bipartite-graph-0067/)

Bipartite graph · qf-mod31-bipartite-odd-cycle · candidate





## 68. An Actual Simple Odd Cycle Certifies Nonbipartiteness

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

View answer and explanation([0, 1, 2], [[0, 1], [1, 2], [0, 2]], 3, Yes, odd_cycle_proves_nonbipartite)

Alternating between two parts around an odd cycle cannot return consistently to the initial part. The certificate requires the full actual cycle, not merely a failed candidate partition or an incomplete walk. The exact result, retaining the requested order, is ([0, 1, 2], [[0, 1], [1, 2], [0, 2]], 3, Yes, odd_cycle_proves_nonbipartite).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r02-bipartite-graph-0068/)

Bipartite graph · qf-mod31-bipartite-odd-cycle · candidate





## 69. An Actual Simple Odd Cycle Certifies Nonbipartiteness

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

View answer and explanation([0, 1, 2], [[0, 1], [1, 2], [0, 2]], 3, Yes, odd_cycle_proves_nonbipartite)

Alternating between two parts around an odd cycle cannot return consistently to the initial part. The certificate requires the full actual cycle, not merely a failed candidate partition or an incomplete walk. The exact result, retaining the requested order, is ([0, 1, 2], [[0, 1], [1, 2], [0, 2]], 3, Yes, odd_cycle_proves_nonbipartite).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r02-bipartite-graph-0069/)

Bipartite graph · qf-mod31-bipartite-odd-cycle · candidate





## 70. An Actual Simple Odd Cycle Certifies Nonbipartiteness

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

View answer and explanation([0, 1, 2], [[0, 1], [1, 2], [0, 2]], 3, Yes, odd_cycle_proves_nonbipartite)

Alternating between two parts around an odd cycle cannot return consistently to the initial part. The certificate requires the full actual cycle, not merely a failed candidate partition or an incomplete walk. The exact result, retaining the requested order, is ([0, 1, 2], [[0, 1], [1, 2], [0, 2]], 3, Yes, odd_cycle_proves_nonbipartite).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r02-bipartite-graph-0070/)

Bipartite graph · qf-mod31-bipartite-odd-cycle · candidate





## 71. An Actual Simple Odd Cycle Certifies Nonbipartiteness

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

View answer and explanation([0, 1, 2], [[0, 1], [1, 2], [0, 2]], 3, Yes, odd_cycle_proves_nonbipartite)

Alternating between two parts around an odd cycle cannot return consistently to the initial part. The certificate requires the full actual cycle, not merely a failed candidate partition or an incomplete walk. The exact result, retaining the requested order, is ([0, 1, 2], [[0, 1], [1, 2], [0, 2]], 3, Yes, odd_cycle_proves_nonbipartite).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r02-bipartite-graph-0071/)

Bipartite graph · qf-mod31-bipartite-odd-cycle · candidate





## 72. An Actual Simple Odd Cycle Certifies Nonbipartiteness

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.

View answer and explanation([0, 1, 2], [[0, 1], [1, 2], [0, 2]], 3, Yes, odd_cycle_proves_nonbipartite)

Alternating between two parts around an odd cycle cannot return consistently to the initial part. The certificate requires the full actual cycle, not merely a failed candidate partition or an incomplete walk. The exact result, retaining the requested order, is ([0, 1, 2], [[0, 1], [1, 2], [0, 2]], 3, Yes, odd_cycle_proves_nonbipartite).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r02-bipartite-graph-0072/)

Bipartite graph · qf-mod31-bipartite-odd-cycle · candidate





## 73. An Actual Simple Odd Cycle Certifies Nonbipartiteness

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], [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.

View answer and explanation([0, 1, 2], [[0, 1], [1, 2], [0, 2]], 3, Yes, odd_cycle_proves_nonbipartite)

Alternating between two parts around an odd cycle cannot return consistently to the initial part. The certificate requires the full actual cycle, not merely a failed candidate partition or an incomplete walk. The exact result, retaining the requested order, is ([0, 1, 2], [[0, 1], [1, 2], [0, 2]], 3, Yes, odd_cycle_proves_nonbipartite).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r02-bipartite-graph-0073/)

Bipartite graph · qf-mod31-bipartite-odd-cycle · candidate





## 74. An Actual Simple Odd Cycle Certifies Nonbipartiteness

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

View answer and explanation([0, 1, 2], [[0, 1], [1, 2], [0, 2]], 3, Yes, odd_cycle_proves_nonbipartite)

Alternating between two parts around an odd cycle cannot return consistently to the initial part. The certificate requires the full actual cycle, not merely a failed candidate partition or an incomplete walk. The exact result, retaining the requested order, is ([0, 1, 2], [[0, 1], [1, 2], [0, 2]], 3, Yes, odd_cycle_proves_nonbipartite).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r02-bipartite-graph-0074/)

Bipartite graph · qf-mod31-bipartite-odd-cycle · candidate





## 75. An Actual Simple Odd Cycle Certifies Nonbipartiteness

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

View answer and explanation([0, 1, 2], [[0, 1], [1, 2], [0, 2]], 3, Yes, odd_cycle_proves_nonbipartite)

Alternating between two parts around an odd cycle cannot return consistently to the initial part. The certificate requires the full actual cycle, not merely a failed candidate partition or an incomplete walk. The exact result, retaining the requested order, is ([0, 1, 2], [[0, 1], [1, 2], [0, 2]], 3, Yes, odd_cycle_proves_nonbipartite).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r02-bipartite-graph-0075/)

Bipartite graph · qf-mod31-bipartite-odd-cycle · candidate





## 76. Isolated Vertices Still Belong to Exactly One Part

Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [2, 3], [3, 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. The total part assignment=[0, 0, 0, 0, 0, 0] uses labels 0 or 1 for every vertex. Its two inverse-image parts are disjoint and cover the whole carrier. Every edge must cross parts; empty parts are allowed in an edgeless graph. Failure of this assignment alone does not prove the graph nonbipartite. vertex=5 must be isolated. Give (vertex,incident_edges,assigned_part,whole_partition_proper?,isolated_vertex_may_be_in_either_part_and_empty_part_allowed_for_edgeless_graph).

View answer and explanation(5, [], 0, No, isolated_vertex_may_be_in_either_part_and_empty_part_allowed_for_edgeless_graph)

Confirm there are no incident edges before reading the part of this vertex. It can be assigned either part without affecting edge constraints, but it must not be omitted from the total assignment. The exact result, retaining the requested order, is (5, [], 0, No, isolated_vertex_may_be_in_either_part_and_empty_part_allowed_for_edgeless_graph).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r02-bipartite-graph-0076/)

Bipartite graph · qf-mod31-bipartite-isolated-boundary · candidate





## 77. Isolated Vertices Still Belong to Exactly One Part

Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [2, 3], [3, 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. The total part assignment=[1, 0, 0, 0, 0, 0] uses labels 0 or 1 for every vertex. Its two inverse-image parts are disjoint and cover the whole carrier. Every edge must cross parts; empty parts are allowed in an edgeless graph. Failure of this assignment alone does not prove the graph nonbipartite. vertex=5 must be isolated. Give (vertex,incident_edges,assigned_part,whole_partition_proper?,isolated_vertex_may_be_in_either_part_and_empty_part_allowed_for_edgeless_graph).

View answer and explanation(5, [], 0, No, isolated_vertex_may_be_in_either_part_and_empty_part_allowed_for_edgeless_graph)

Confirm there are no incident edges before reading the part of this vertex. It can be assigned either part without affecting edge constraints, but it must not be omitted from the total assignment. The exact result, retaining the requested order, is (5, [], 0, No, isolated_vertex_may_be_in_either_part_and_empty_part_allowed_for_edgeless_graph).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r02-bipartite-graph-0077/)

Bipartite graph · qf-mod31-bipartite-isolated-boundary · candidate





## 78. Isolated Vertices Still Belong to Exactly One Part

Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [2, 3], [3, 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. The total part assignment=[0, 1, 0, 0, 0, 0] uses labels 0 or 1 for every vertex. Its two inverse-image parts are disjoint and cover the whole carrier. Every edge must cross parts; empty parts are allowed in an edgeless graph. Failure of this assignment alone does not prove the graph nonbipartite. vertex=5 must be isolated. Give (vertex,incident_edges,assigned_part,whole_partition_proper?,isolated_vertex_may_be_in_either_part_and_empty_part_allowed_for_edgeless_graph).

View answer and explanation(5, [], 0, No, isolated_vertex_may_be_in_either_part_and_empty_part_allowed_for_edgeless_graph)

Confirm there are no incident edges before reading the part of this vertex. It can be assigned either part without affecting edge constraints, but it must not be omitted from the total assignment. The exact result, retaining the requested order, is (5, [], 0, No, isolated_vertex_may_be_in_either_part_and_empty_part_allowed_for_edgeless_graph).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r02-bipartite-graph-0078/)

Bipartite graph · qf-mod31-bipartite-isolated-boundary · candidate





## 79. Isolated Vertices Still Belong to Exactly One Part

Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [2, 3], [3, 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. The total part assignment=[1, 1, 0, 0, 0, 0] uses labels 0 or 1 for every vertex. Its two inverse-image parts are disjoint and cover the whole carrier. Every edge must cross parts; empty parts are allowed in an edgeless graph. Failure of this assignment alone does not prove the graph nonbipartite. vertex=5 must be isolated. Give (vertex,incident_edges,assigned_part,whole_partition_proper?,isolated_vertex_may_be_in_either_part_and_empty_part_allowed_for_edgeless_graph).

View answer and explanation(5, [], 0, No, isolated_vertex_may_be_in_either_part_and_empty_part_allowed_for_edgeless_graph)

Confirm there are no incident edges before reading the part of this vertex. It can be assigned either part without affecting edge constraints, but it must not be omitted from the total assignment. The exact result, retaining the requested order, is (5, [], 0, No, isolated_vertex_may_be_in_either_part_and_empty_part_allowed_for_edgeless_graph).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r02-bipartite-graph-0079/)

Bipartite graph · qf-mod31-bipartite-isolated-boundary · candidate





## 80. Isolated Vertices Still Belong to Exactly One Part

Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [2, 3], [3, 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. The total part assignment=[0, 0, 1, 0, 0, 0] uses labels 0 or 1 for every vertex. Its two inverse-image parts are disjoint and cover the whole carrier. Every edge must cross parts; empty parts are allowed in an edgeless graph. Failure of this assignment alone does not prove the graph nonbipartite. vertex=5 must be isolated. Give (vertex,incident_edges,assigned_part,whole_partition_proper?,isolated_vertex_may_be_in_either_part_and_empty_part_allowed_for_edgeless_graph).

View answer and explanation(5, [], 0, No, isolated_vertex_may_be_in_either_part_and_empty_part_allowed_for_edgeless_graph)

Confirm there are no incident edges before reading the part of this vertex. It can be assigned either part without affecting edge constraints, but it must not be omitted from the total assignment. The exact result, retaining the requested order, is (5, [], 0, No, isolated_vertex_may_be_in_either_part_and_empty_part_allowed_for_edgeless_graph).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r02-bipartite-graph-0080/)

Bipartite graph · qf-mod31-bipartite-isolated-boundary · candidate





## 81. Isolated Vertices Still Belong to Exactly One Part

Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [2, 3], [3, 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. The total part assignment=[1, 0, 1, 0, 0, 0] uses labels 0 or 1 for every vertex. Its two inverse-image parts are disjoint and cover the whole carrier. Every edge must cross parts; empty parts are allowed in an edgeless graph. Failure of this assignment alone does not prove the graph nonbipartite. vertex=5 must be isolated. Give (vertex,incident_edges,assigned_part,whole_partition_proper?,isolated_vertex_may_be_in_either_part_and_empty_part_allowed_for_edgeless_graph).

View answer and explanation(5, [], 0, No, isolated_vertex_may_be_in_either_part_and_empty_part_allowed_for_edgeless_graph)

Confirm there are no incident edges before reading the part of this vertex. It can be assigned either part without affecting edge constraints, but it must not be omitted from the total assignment. The exact result, retaining the requested order, is (5, [], 0, No, isolated_vertex_may_be_in_either_part_and_empty_part_allowed_for_edgeless_graph).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r02-bipartite-graph-0081/)

Bipartite graph · qf-mod31-bipartite-isolated-boundary · candidate





## 82. Isolated Vertices Still Belong to Exactly One Part

Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [2, 3], [3, 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. The total part assignment=[0, 1, 1, 0, 0, 0] uses labels 0 or 1 for every vertex. Its two inverse-image parts are disjoint and cover the whole carrier. Every edge must cross parts; empty parts are allowed in an edgeless graph. Failure of this assignment alone does not prove the graph nonbipartite. vertex=5 must be isolated. Give (vertex,incident_edges,assigned_part,whole_partition_proper?,isolated_vertex_may_be_in_either_part_and_empty_part_allowed_for_edgeless_graph).

View answer and explanation(5, [], 0, No, isolated_vertex_may_be_in_either_part_and_empty_part_allowed_for_edgeless_graph)

Confirm there are no incident edges before reading the part of this vertex. It can be assigned either part without affecting edge constraints, but it must not be omitted from the total assignment. The exact result, retaining the requested order, is (5, [], 0, No, isolated_vertex_may_be_in_either_part_and_empty_part_allowed_for_edgeless_graph).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r02-bipartite-graph-0082/)

Bipartite graph · qf-mod31-bipartite-isolated-boundary · candidate





## 83. Isolated Vertices Still Belong to Exactly One Part

Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [2, 3], [3, 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. The total part assignment=[1, 1, 1, 0, 0, 0] uses labels 0 or 1 for every vertex. Its two inverse-image parts are disjoint and cover the whole carrier. Every edge must cross parts; empty parts are allowed in an edgeless graph. Failure of this assignment alone does not prove the graph nonbipartite. vertex=5 must be isolated. Give (vertex,incident_edges,assigned_part,whole_partition_proper?,isolated_vertex_may_be_in_either_part_and_empty_part_allowed_for_edgeless_graph).

View answer and explanation(5, [], 0, No, isolated_vertex_may_be_in_either_part_and_empty_part_allowed_for_edgeless_graph)

Confirm there are no incident edges before reading the part of this vertex. It can be assigned either part without affecting edge constraints, but it must not be omitted from the total assignment. The exact result, retaining the requested order, is (5, [], 0, No, isolated_vertex_may_be_in_either_part_and_empty_part_allowed_for_edgeless_graph).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r02-bipartite-graph-0083/)

Bipartite graph · qf-mod31-bipartite-isolated-boundary · candidate





## 84. Isolated Vertices Still Belong to Exactly One Part

Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [2, 3], [3, 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. The total part assignment=[0, 0, 0, 1, 0, 0] uses labels 0 or 1 for every vertex. Its two inverse-image parts are disjoint and cover the whole carrier. Every edge must cross parts; empty parts are allowed in an edgeless graph. Failure of this assignment alone does not prove the graph nonbipartite. vertex=5 must be isolated. Give (vertex,incident_edges,assigned_part,whole_partition_proper?,isolated_vertex_may_be_in_either_part_and_empty_part_allowed_for_edgeless_graph).

View answer and explanation(5, [], 0, No, isolated_vertex_may_be_in_either_part_and_empty_part_allowed_for_edgeless_graph)

Confirm there are no incident edges before reading the part of this vertex. It can be assigned either part without affecting edge constraints, but it must not be omitted from the total assignment. The exact result, retaining the requested order, is (5, [], 0, No, isolated_vertex_may_be_in_either_part_and_empty_part_allowed_for_edgeless_graph).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r02-bipartite-graph-0084/)

Bipartite graph · qf-mod31-bipartite-isolated-boundary · candidate





## 85. Isolated Vertices Still Belong to Exactly One Part

Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [2, 3], [3, 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. The total part assignment=[1, 0, 0, 1, 0, 0] uses labels 0 or 1 for every vertex. Its two inverse-image parts are disjoint and cover the whole carrier. Every edge must cross parts; empty parts are allowed in an edgeless graph. Failure of this assignment alone does not prove the graph nonbipartite. vertex=5 must be isolated. Give (vertex,incident_edges,assigned_part,whole_partition_proper?,isolated_vertex_may_be_in_either_part_and_empty_part_allowed_for_edgeless_graph).

View answer and explanation(5, [], 0, No, isolated_vertex_may_be_in_either_part_and_empty_part_allowed_for_edgeless_graph)

Confirm there are no incident edges before reading the part of this vertex. It can be assigned either part without affecting edge constraints, but it must not be omitted from the total assignment. The exact result, retaining the requested order, is (5, [], 0, No, isolated_vertex_may_be_in_either_part_and_empty_part_allowed_for_edgeless_graph).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r02-bipartite-graph-0085/)

Bipartite graph · qf-mod31-bipartite-isolated-boundary · candidate





## 86. Isolated Vertices Still Belong to Exactly One Part

Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [2, 3], [3, 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. The total part assignment=[0, 1, 0, 1, 0, 0] uses labels 0 or 1 for every vertex. Its two inverse-image parts are disjoint and cover the whole carrier. Every edge must cross parts; empty parts are allowed in an edgeless graph. Failure of this assignment alone does not prove the graph nonbipartite. vertex=5 must be isolated. Give (vertex,incident_edges,assigned_part,whole_partition_proper?,isolated_vertex_may_be_in_either_part_and_empty_part_allowed_for_edgeless_graph).

View answer and explanation(5, [], 0, Yes, isolated_vertex_may_be_in_either_part_and_empty_part_allowed_for_edgeless_graph)

Confirm there are no incident edges before reading the part of this vertex. It can be assigned either part without affecting edge constraints, but it must not be omitted from the total assignment. The exact result, retaining the requested order, is (5, [], 0, Yes, isolated_vertex_may_be_in_either_part_and_empty_part_allowed_for_edgeless_graph).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r02-bipartite-graph-0086/)

Bipartite graph · qf-mod31-bipartite-isolated-boundary · candidate





## 87. Isolated Vertices Still Belong to Exactly One Part

Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [2, 3], [3, 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. The total part assignment=[1, 1, 0, 1, 0, 0] uses labels 0 or 1 for every vertex. Its two inverse-image parts are disjoint and cover the whole carrier. Every edge must cross parts; empty parts are allowed in an edgeless graph. Failure of this assignment alone does not prove the graph nonbipartite. vertex=5 must be isolated. Give (vertex,incident_edges,assigned_part,whole_partition_proper?,isolated_vertex_may_be_in_either_part_and_empty_part_allowed_for_edgeless_graph).

View answer and explanation(5, [], 0, No, isolated_vertex_may_be_in_either_part_and_empty_part_allowed_for_edgeless_graph)

Confirm there are no incident edges before reading the part of this vertex. It can be assigned either part without affecting edge constraints, but it must not be omitted from the total assignment. The exact result, retaining the requested order, is (5, [], 0, No, isolated_vertex_may_be_in_either_part_and_empty_part_allowed_for_edgeless_graph).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r02-bipartite-graph-0087/)

Bipartite graph · qf-mod31-bipartite-isolated-boundary · candidate





## 88. Isolated Vertices Still Belong to Exactly One Part

Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [2, 3], [3, 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. The total part assignment=[0, 0, 1, 1, 0, 0] uses labels 0 or 1 for every vertex. Its two inverse-image parts are disjoint and cover the whole carrier. Every edge must cross parts; empty parts are allowed in an edgeless graph. Failure of this assignment alone does not prove the graph nonbipartite. vertex=5 must be isolated. Give (vertex,incident_edges,assigned_part,whole_partition_proper?,isolated_vertex_may_be_in_either_part_and_empty_part_allowed_for_edgeless_graph).

View answer and explanation(5, [], 0, No, isolated_vertex_may_be_in_either_part_and_empty_part_allowed_for_edgeless_graph)

Confirm there are no incident edges before reading the part of this vertex. It can be assigned either part without affecting edge constraints, but it must not be omitted from the total assignment. The exact result, retaining the requested order, is (5, [], 0, No, isolated_vertex_may_be_in_either_part_and_empty_part_allowed_for_edgeless_graph).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r02-bipartite-graph-0088/)

Bipartite graph · qf-mod31-bipartite-isolated-boundary · candidate





## 89. Isolated Vertices Still Belong to Exactly One Part

Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [2, 3], [3, 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. The total part assignment=[1, 0, 1, 1, 0, 0] uses labels 0 or 1 for every vertex. Its two inverse-image parts are disjoint and cover the whole carrier. Every edge must cross parts; empty parts are allowed in an edgeless graph. Failure of this assignment alone does not prove the graph nonbipartite. vertex=5 must be isolated. Give (vertex,incident_edges,assigned_part,whole_partition_proper?,isolated_vertex_may_be_in_either_part_and_empty_part_allowed_for_edgeless_graph).

View answer and explanation(5, [], 0, No, isolated_vertex_may_be_in_either_part_and_empty_part_allowed_for_edgeless_graph)

Confirm there are no incident edges before reading the part of this vertex. It can be assigned either part without affecting edge constraints, but it must not be omitted from the total assignment. The exact result, retaining the requested order, is (5, [], 0, No, isolated_vertex_may_be_in_either_part_and_empty_part_allowed_for_edgeless_graph).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r02-bipartite-graph-0089/)

Bipartite graph · qf-mod31-bipartite-isolated-boundary · candidate





## 90. Isolated Vertices Still Belong to Exactly One Part

Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [2, 3], [3, 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. The total part assignment=[0, 1, 1, 1, 0, 0] uses labels 0 or 1 for every vertex. Its two inverse-image parts are disjoint and cover the whole carrier. Every edge must cross parts; empty parts are allowed in an edgeless graph. Failure of this assignment alone does not prove the graph nonbipartite. vertex=5 must be isolated. Give (vertex,incident_edges,assigned_part,whole_partition_proper?,isolated_vertex_may_be_in_either_part_and_empty_part_allowed_for_edgeless_graph).

View answer and explanation(5, [], 0, No, isolated_vertex_may_be_in_either_part_and_empty_part_allowed_for_edgeless_graph)

Confirm there are no incident edges before reading the part of this vertex. It can be assigned either part without affecting edge constraints, but it must not be omitted from the total assignment. The exact result, retaining the requested order, is (5, [], 0, No, isolated_vertex_may_be_in_either_part_and_empty_part_allowed_for_edgeless_graph).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r02-bipartite-graph-0090/)

Bipartite graph · qf-mod31-bipartite-isolated-boundary · candidate





## 91. Isolated Vertices Still Belong to Exactly One Part

Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [2, 3], [3, 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. The total part assignment=[1, 1, 1, 1, 0, 0] uses labels 0 or 1 for every vertex. Its two inverse-image parts are disjoint and cover the whole carrier. Every edge must cross parts; empty parts are allowed in an edgeless graph. Failure of this assignment alone does not prove the graph nonbipartite. vertex=5 must be isolated. Give (vertex,incident_edges,assigned_part,whole_partition_proper?,isolated_vertex_may_be_in_either_part_and_empty_part_allowed_for_edgeless_graph).

View answer and explanation(5, [], 0, No, isolated_vertex_may_be_in_either_part_and_empty_part_allowed_for_edgeless_graph)

Confirm there are no incident edges before reading the part of this vertex. It can be assigned either part without affecting edge constraints, but it must not be omitted from the total assignment. The exact result, retaining the requested order, is (5, [], 0, No, isolated_vertex_may_be_in_either_part_and_empty_part_allowed_for_edgeless_graph).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r02-bipartite-graph-0091/)

Bipartite graph · qf-mod31-bipartite-isolated-boundary · candidate





## 92. Isolated Vertices Still Belong to Exactly One Part

Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [2, 3], [3, 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. The total part assignment=[0, 0, 0, 0, 1, 0] uses labels 0 or 1 for every vertex. Its two inverse-image parts are disjoint and cover the whole carrier. Every edge must cross parts; empty parts are allowed in an edgeless graph. Failure of this assignment alone does not prove the graph nonbipartite. vertex=5 must be isolated. Give (vertex,incident_edges,assigned_part,whole_partition_proper?,isolated_vertex_may_be_in_either_part_and_empty_part_allowed_for_edgeless_graph).

View answer and explanation(5, [], 0, No, isolated_vertex_may_be_in_either_part_and_empty_part_allowed_for_edgeless_graph)

Confirm there are no incident edges before reading the part of this vertex. It can be assigned either part without affecting edge constraints, but it must not be omitted from the total assignment. The exact result, retaining the requested order, is (5, [], 0, No, isolated_vertex_may_be_in_either_part_and_empty_part_allowed_for_edgeless_graph).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r02-bipartite-graph-0092/)

Bipartite graph · qf-mod31-bipartite-isolated-boundary · candidate





## 93. Isolated Vertices Still Belong to Exactly One Part

Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [2, 3], [3, 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. The total part assignment=[1, 0, 0, 0, 1, 0] uses labels 0 or 1 for every vertex. Its two inverse-image parts are disjoint and cover the whole carrier. Every edge must cross parts; empty parts are allowed in an edgeless graph. Failure of this assignment alone does not prove the graph nonbipartite. vertex=5 must be isolated. Give (vertex,incident_edges,assigned_part,whole_partition_proper?,isolated_vertex_may_be_in_either_part_and_empty_part_allowed_for_edgeless_graph).

View answer and explanation(5, [], 0, No, isolated_vertex_may_be_in_either_part_and_empty_part_allowed_for_edgeless_graph)

Confirm there are no incident edges before reading the part of this vertex. It can be assigned either part without affecting edge constraints, but it must not be omitted from the total assignment. The exact result, retaining the requested order, is (5, [], 0, No, isolated_vertex_may_be_in_either_part_and_empty_part_allowed_for_edgeless_graph).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r02-bipartite-graph-0093/)

Bipartite graph · qf-mod31-bipartite-isolated-boundary · candidate





## 94. Isolated Vertices Still Belong to Exactly One Part

Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [2, 3], [3, 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. The total part assignment=[0, 1, 0, 0, 1, 0] uses labels 0 or 1 for every vertex. Its two inverse-image parts are disjoint and cover the whole carrier. Every edge must cross parts; empty parts are allowed in an edgeless graph. Failure of this assignment alone does not prove the graph nonbipartite. vertex=5 must be isolated. Give (vertex,incident_edges,assigned_part,whole_partition_proper?,isolated_vertex_may_be_in_either_part_and_empty_part_allowed_for_edgeless_graph).

View answer and explanation(5, [], 0, No, isolated_vertex_may_be_in_either_part_and_empty_part_allowed_for_edgeless_graph)

Confirm there are no incident edges before reading the part of this vertex. It can be assigned either part without affecting edge constraints, but it must not be omitted from the total assignment. The exact result, retaining the requested order, is (5, [], 0, No, isolated_vertex_may_be_in_either_part_and_empty_part_allowed_for_edgeless_graph).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r02-bipartite-graph-0094/)

Bipartite graph · qf-mod31-bipartite-isolated-boundary · candidate





## 95. Isolated Vertices Still Belong to Exactly One Part

Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [2, 3], [3, 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. The total part assignment=[1, 1, 0, 0, 1, 0] uses labels 0 or 1 for every vertex. Its two inverse-image parts are disjoint and cover the whole carrier. Every edge must cross parts; empty parts are allowed in an edgeless graph. Failure of this assignment alone does not prove the graph nonbipartite. vertex=5 must be isolated. Give (vertex,incident_edges,assigned_part,whole_partition_proper?,isolated_vertex_may_be_in_either_part_and_empty_part_allowed_for_edgeless_graph).

View answer and explanation(5, [], 0, No, isolated_vertex_may_be_in_either_part_and_empty_part_allowed_for_edgeless_graph)

Confirm there are no incident edges before reading the part of this vertex. It can be assigned either part without affecting edge constraints, but it must not be omitted from the total assignment. The exact result, retaining the requested order, is (5, [], 0, No, isolated_vertex_may_be_in_either_part_and_empty_part_allowed_for_edgeless_graph).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r02-bipartite-graph-0095/)

Bipartite graph · qf-mod31-bipartite-isolated-boundary · candidate





## 96. Isolated Vertices Still Belong to Exactly One Part

Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [2, 3], [3, 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. The total part assignment=[0, 0, 1, 0, 1, 0] uses labels 0 or 1 for every vertex. Its two inverse-image parts are disjoint and cover the whole carrier. Every edge must cross parts; empty parts are allowed in an edgeless graph. Failure of this assignment alone does not prove the graph nonbipartite. vertex=5 must be isolated. Give (vertex,incident_edges,assigned_part,whole_partition_proper?,isolated_vertex_may_be_in_either_part_and_empty_part_allowed_for_edgeless_graph).

View answer and explanation(5, [], 0, No, isolated_vertex_may_be_in_either_part_and_empty_part_allowed_for_edgeless_graph)

Confirm there are no incident edges before reading the part of this vertex. It can be assigned either part without affecting edge constraints, but it must not be omitted from the total assignment. The exact result, retaining the requested order, is (5, [], 0, No, isolated_vertex_may_be_in_either_part_and_empty_part_allowed_for_edgeless_graph).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r02-bipartite-graph-0096/)

Bipartite graph · qf-mod31-bipartite-isolated-boundary · candidate





## 97. Isolated Vertices Still Belong to Exactly One Part

Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [2, 3], [3, 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. The total part assignment=[1, 0, 1, 0, 1, 0] uses labels 0 or 1 for every vertex. Its two inverse-image parts are disjoint and cover the whole carrier. Every edge must cross parts; empty parts are allowed in an edgeless graph. Failure of this assignment alone does not prove the graph nonbipartite. vertex=5 must be isolated. Give (vertex,incident_edges,assigned_part,whole_partition_proper?,isolated_vertex_may_be_in_either_part_and_empty_part_allowed_for_edgeless_graph).

View answer and explanation(5, [], 0, Yes, isolated_vertex_may_be_in_either_part_and_empty_part_allowed_for_edgeless_graph)

Confirm there are no incident edges before reading the part of this vertex. It can be assigned either part without affecting edge constraints, but it must not be omitted from the total assignment. The exact result, retaining the requested order, is (5, [], 0, Yes, isolated_vertex_may_be_in_either_part_and_empty_part_allowed_for_edgeless_graph).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r02-bipartite-graph-0097/)

Bipartite graph · qf-mod31-bipartite-isolated-boundary · candidate





## 98. Isolated Vertices Still Belong to Exactly One Part

Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [2, 3], [3, 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. The total part assignment=[0, 1, 1, 0, 1, 0] uses labels 0 or 1 for every vertex. Its two inverse-image parts are disjoint and cover the whole carrier. Every edge must cross parts; empty parts are allowed in an edgeless graph. Failure of this assignment alone does not prove the graph nonbipartite. vertex=5 must be isolated. Give (vertex,incident_edges,assigned_part,whole_partition_proper?,isolated_vertex_may_be_in_either_part_and_empty_part_allowed_for_edgeless_graph).

View answer and explanation(5, [], 0, No, isolated_vertex_may_be_in_either_part_and_empty_part_allowed_for_edgeless_graph)

Confirm there are no incident edges before reading the part of this vertex. It can be assigned either part without affecting edge constraints, but it must not be omitted from the total assignment. The exact result, retaining the requested order, is (5, [], 0, No, isolated_vertex_may_be_in_either_part_and_empty_part_allowed_for_edgeless_graph).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r02-bipartite-graph-0098/)

Bipartite graph · qf-mod31-bipartite-isolated-boundary · candidate





## 99. Isolated Vertices Still Belong to Exactly One Part

Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [2, 3], [3, 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. The total part assignment=[1, 1, 1, 0, 1, 0] uses labels 0 or 1 for every vertex. Its two inverse-image parts are disjoint and cover the whole carrier. Every edge must cross parts; empty parts are allowed in an edgeless graph. Failure of this assignment alone does not prove the graph nonbipartite. vertex=5 must be isolated. Give (vertex,incident_edges,assigned_part,whole_partition_proper?,isolated_vertex_may_be_in_either_part_and_empty_part_allowed_for_edgeless_graph).

View answer and explanation(5, [], 0, No, isolated_vertex_may_be_in_either_part_and_empty_part_allowed_for_edgeless_graph)

Confirm there are no incident edges before reading the part of this vertex. It can be assigned either part without affecting edge constraints, but it must not be omitted from the total assignment. The exact result, retaining the requested order, is (5, [], 0, No, isolated_vertex_may_be_in_either_part_and_empty_part_allowed_for_edgeless_graph).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r02-bipartite-graph-0099/)

Bipartite graph · qf-mod31-bipartite-isolated-boundary · candidate





## 100. Isolated Vertices Still Belong to Exactly One Part

Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edges=[[0, 1], [1, 2], [2, 3], [3, 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. The total part assignment=[0, 0, 0, 1, 1, 0] uses labels 0 or 1 for every vertex. Its two inverse-image parts are disjoint and cover the whole carrier. Every edge must cross parts; empty parts are allowed in an edgeless graph. Failure of this assignment alone does not prove the graph nonbipartite. vertex=5 must be isolated. Give (vertex,incident_edges,assigned_part,whole_partition_proper?,isolated_vertex_may_be_in_either_part_and_empty_part_allowed_for_edgeless_graph).

View answer and explanation(5, [], 0, No, isolated_vertex_may_be_in_either_part_and_empty_part_allowed_for_edgeless_graph)

Confirm there are no incident edges before reading the part of this vertex. It can be assigned either part without affecting edge constraints, but it must not be omitted from the total assignment. The exact result, retaining the requested order, is (5, [], 0, No, isolated_vertex_may_be_in_either_part_and_empty_part_allowed_for_edgeless_graph).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r02-bipartite-graph-0100/)

Bipartite graph · qf-mod31-bipartite-isolated-boundary · candidate



[Canonical question JSONL](https://ocme.evemisslab.com/data/questions/batches/batch-20261014-advance-20261031-r02-bipartite-graph-foundations/questions.jsonl) · [English JSONL](https://ocme.evemisslab.com/data/i18n/questions/batch-20261014-advance-20261031-r02-bipartite-graph-foundations.en.jsonl) · [Validation report](https://ocme.evemisslab.com/data/questions/batches/batch-20261014-advance-20261031-r02-bipartite-graph-foundations/validation-report.json)