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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
Bipartite graph · qf-mod31-bipartite-isolated-boundary · candidate
Canonical question JSONL · English JSONL · Validation report