# Endpoint-Disjoint Matchings, Maximality and Larger-Witness Distinctions

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-20261101-r05-graph-matching-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. Selected Edges Form a Matching Only When Endpoints Are Disjoint

Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [2, 3]], endpoints increasing and the edge list lexicographically sorted. The source rejects noncanonical or duplicate edges. Select original edge indexes matching=[0, 1], sorted increasingly with duplicates forbidden. A matching has pairwise disjoint endpoints; the empty matching is allowed. Keep matching validity, maximality, maximum cardinality and perfect coverage distinct. Give (matching_indexes,matching_edges,conflicting_vertices,valid_matching?). List conflicting vertices in first-seen endpoint order while traversing the selected edges, not an invented numerical sort.

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

Read every selected original edge and count endpoint incidences. Any count above one is a conflict. Validity alone supplies no maximum-cardinality or perfect-matching certificate. The exact result, retaining the requested order, is ([0, 1], [[0, 1], [2, 3]], [], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r05-graph-matching-0001/)

Graph matching · qf-mod32-matching-audit · candidate





## 2. Selected Edges Form a Matching Only When Endpoints Are Disjoint

Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [2, 3]], endpoints increasing and the edge list lexicographically sorted. The source rejects noncanonical or duplicate edges. Select original edge indexes matching=[0, 2], sorted increasingly with duplicates forbidden. A matching has pairwise disjoint endpoints; the empty matching is allowed. Keep matching validity, maximality, maximum cardinality and perfect coverage distinct. Give (matching_indexes,matching_edges,conflicting_vertices,valid_matching?). List conflicting vertices in first-seen endpoint order while traversing the selected edges, not an invented numerical sort.

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

Read every selected original edge and count endpoint incidences. Any count above one is a conflict. Validity alone supplies no maximum-cardinality or perfect-matching certificate. The exact result, retaining the requested order, is ([0, 2], [[0, 1], [2, 3]], [], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r05-graph-matching-0002/)

Graph matching · qf-mod32-matching-audit · candidate





## 3. Selected Edges Form a Matching Only When Endpoints Are Disjoint

Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 3], [2, 3]], endpoints increasing and the edge list lexicographically sorted. The source rejects noncanonical or duplicate edges. Select original edge indexes matching=[0, 2], sorted increasingly with duplicates forbidden. A matching has pairwise disjoint endpoints; the empty matching is allowed. Keep matching validity, maximality, maximum cardinality and perfect coverage distinct. Give (matching_indexes,matching_edges,conflicting_vertices,valid_matching?). List conflicting vertices in first-seen endpoint order while traversing the selected edges, not an invented numerical sort.

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

Read every selected original edge and count endpoint incidences. Any count above one is a conflict. Validity alone supplies no maximum-cardinality or perfect-matching certificate. The exact result, retaining the requested order, is ([0, 2], [[0, 1], [2, 3]], [], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r05-graph-matching-0003/)

Graph matching · qf-mod32-matching-audit · candidate





## 4. Selected Edges Form a Matching Only When Endpoints Are Disjoint

Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 3], [2, 3]], endpoints increasing and the edge list lexicographically sorted. The source rejects noncanonical or duplicate edges. Select original edge indexes matching=[0, 3], sorted increasingly with duplicates forbidden. A matching has pairwise disjoint endpoints; the empty matching is allowed. Keep matching validity, maximality, maximum cardinality and perfect coverage distinct. Give (matching_indexes,matching_edges,conflicting_vertices,valid_matching?). List conflicting vertices in first-seen endpoint order while traversing the selected edges, not an invented numerical sort.

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

Read every selected original edge and count endpoint incidences. Any count above one is a conflict. Validity alone supplies no maximum-cardinality or perfect-matching certificate. The exact result, retaining the requested order, is ([0, 3], [[0, 1], [2, 3]], [], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r05-graph-matching-0004/)

Graph matching · qf-mod32-matching-audit · candidate





## 5. Selected Edges Form a Matching Only When Endpoints Are Disjoint

Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [1, 2], [2, 3]], endpoints increasing and the edge list lexicographically sorted. The source rejects noncanonical or duplicate edges. Select original edge indexes matching=[0, 2], sorted increasingly with duplicates forbidden. A matching has pairwise disjoint endpoints; the empty matching is allowed. Keep matching validity, maximality, maximum cardinality and perfect coverage distinct. Give (matching_indexes,matching_edges,conflicting_vertices,valid_matching?). List conflicting vertices in first-seen endpoint order while traversing the selected edges, not an invented numerical sort.

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

Read every selected original edge and count endpoint incidences. Any count above one is a conflict. Validity alone supplies no maximum-cardinality or perfect-matching certificate. The exact result, retaining the requested order, is ([0, 2], [[0, 1], [2, 3]], [], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r05-graph-matching-0005/)

Graph matching · qf-mod32-matching-audit · candidate





## 6. Selected Edges Form a Matching Only When Endpoints Are Disjoint

Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [1, 2], [2, 3]], endpoints increasing and the edge list lexicographically sorted. The source rejects noncanonical or duplicate edges. Select original edge indexes matching=[0, 3], sorted increasingly with duplicates forbidden. A matching has pairwise disjoint endpoints; the empty matching is allowed. Keep matching validity, maximality, maximum cardinality and perfect coverage distinct. Give (matching_indexes,matching_edges,conflicting_vertices,valid_matching?). List conflicting vertices in first-seen endpoint order while traversing the selected edges, not an invented numerical sort.

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

Read every selected original edge and count endpoint incidences. Any count above one is a conflict. Validity alone supplies no maximum-cardinality or perfect-matching certificate. The exact result, retaining the requested order, is ([0, 3], [[0, 1], [2, 3]], [], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r05-graph-matching-0006/)

Graph matching · qf-mod32-matching-audit · candidate





## 7. Selected Edges Form a Matching Only When Endpoints Are Disjoint

Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 3], [1, 2], [2, 3]], endpoints increasing and the edge list lexicographically sorted. The source rejects noncanonical or duplicate edges. Select original edge indexes matching=[0, 3], sorted increasingly with duplicates forbidden. A matching has pairwise disjoint endpoints; the empty matching is allowed. Keep matching validity, maximality, maximum cardinality and perfect coverage distinct. Give (matching_indexes,matching_edges,conflicting_vertices,valid_matching?). List conflicting vertices in first-seen endpoint order while traversing the selected edges, not an invented numerical sort.

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

Read every selected original edge and count endpoint incidences. Any count above one is a conflict. Validity alone supplies no maximum-cardinality or perfect-matching certificate. The exact result, retaining the requested order, is ([0, 3], [[0, 1], [2, 3]], [], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r05-graph-matching-0007/)

Graph matching · qf-mod32-matching-audit · candidate





## 8. Selected Edges Form a Matching Only When Endpoints Are Disjoint

Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 3], [1, 2], [2, 3]], endpoints increasing and the edge list lexicographically sorted. The source rejects noncanonical or duplicate edges. Select original edge indexes matching=[0, 4], sorted increasingly with duplicates forbidden. A matching has pairwise disjoint endpoints; the empty matching is allowed. Keep matching validity, maximality, maximum cardinality and perfect coverage distinct. Give (matching_indexes,matching_edges,conflicting_vertices,valid_matching?). List conflicting vertices in first-seen endpoint order while traversing the selected edges, not an invented numerical sort.

View answer and explanation([0, 4], [[0, 1], [2, 3]], [], Yes)

Read every selected original edge and count endpoint incidences. Any count above one is a conflict. Validity alone supplies no maximum-cardinality or perfect-matching certificate. The exact result, retaining the requested order, is ([0, 4], [[0, 1], [2, 3]], [], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r05-graph-matching-0008/)

Graph matching · qf-mod32-matching-audit · candidate





## 9. Selected Edges Form a Matching Only When Endpoints Are Disjoint

Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [1, 3], [2, 3]], endpoints increasing and the edge list lexicographically sorted. The source rejects noncanonical or duplicate edges. Select original edge indexes matching=[0, 2], sorted increasingly with duplicates forbidden. A matching has pairwise disjoint endpoints; the empty matching is allowed. Keep matching validity, maximality, maximum cardinality and perfect coverage distinct. Give (matching_indexes,matching_edges,conflicting_vertices,valid_matching?). List conflicting vertices in first-seen endpoint order while traversing the selected edges, not an invented numerical sort.

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

Read every selected original edge and count endpoint incidences. Any count above one is a conflict. Validity alone supplies no maximum-cardinality or perfect-matching certificate. The exact result, retaining the requested order, is ([0, 2], [[0, 1], [2, 3]], [], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r05-graph-matching-0009/)

Graph matching · qf-mod32-matching-audit · candidate





## 10. Selected Edges Form a Matching Only When Endpoints Are Disjoint

Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [1, 3], [2, 3]], endpoints increasing and the edge list lexicographically sorted. The source rejects noncanonical or duplicate edges. Select original edge indexes matching=[0, 3], sorted increasingly with duplicates forbidden. A matching has pairwise disjoint endpoints; the empty matching is allowed. Keep matching validity, maximality, maximum cardinality and perfect coverage distinct. Give (matching_indexes,matching_edges,conflicting_vertices,valid_matching?). List conflicting vertices in first-seen endpoint order while traversing the selected edges, not an invented numerical sort.

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

Read every selected original edge and count endpoint incidences. Any count above one is a conflict. Validity alone supplies no maximum-cardinality or perfect-matching certificate. The exact result, retaining the requested order, is ([0, 3], [[0, 1], [2, 3]], [], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r05-graph-matching-0010/)

Graph matching · qf-mod32-matching-audit · candidate





## 11. Selected Edges Form a Matching Only When Endpoints Are Disjoint

Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 3], [1, 3], [2, 3]], endpoints increasing and the edge list lexicographically sorted. The source rejects noncanonical or duplicate edges. Select original edge indexes matching=[0, 3], sorted increasingly with duplicates forbidden. A matching has pairwise disjoint endpoints; the empty matching is allowed. Keep matching validity, maximality, maximum cardinality and perfect coverage distinct. Give (matching_indexes,matching_edges,conflicting_vertices,valid_matching?). List conflicting vertices in first-seen endpoint order while traversing the selected edges, not an invented numerical sort.

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

Read every selected original edge and count endpoint incidences. Any count above one is a conflict. Validity alone supplies no maximum-cardinality or perfect-matching certificate. The exact result, retaining the requested order, is ([0, 3], [[0, 1], [2, 3]], [], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r05-graph-matching-0011/)

Graph matching · qf-mod32-matching-audit · candidate





## 12. Selected Edges Form a Matching Only When Endpoints Are Disjoint

Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 3], [1, 3], [2, 3]], endpoints increasing and the edge list lexicographically sorted. The source rejects noncanonical or duplicate edges. Select original edge indexes matching=[0, 4], sorted increasingly with duplicates forbidden. A matching has pairwise disjoint endpoints; the empty matching is allowed. Keep matching validity, maximality, maximum cardinality and perfect coverage distinct. Give (matching_indexes,matching_edges,conflicting_vertices,valid_matching?). List conflicting vertices in first-seen endpoint order while traversing the selected edges, not an invented numerical sort.

View answer and explanation([0, 4], [[0, 1], [2, 3]], [], Yes)

Read every selected original edge and count endpoint incidences. Any count above one is a conflict. Validity alone supplies no maximum-cardinality or perfect-matching certificate. The exact result, retaining the requested order, is ([0, 4], [[0, 1], [2, 3]], [], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r05-graph-matching-0012/)

Graph matching · qf-mod32-matching-audit · candidate





## 13. Selected Edges Form a Matching Only When Endpoints Are Disjoint

Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [1, 2], [1, 3], [2, 3]], endpoints increasing and the edge list lexicographically sorted. The source rejects noncanonical or duplicate edges. Select original edge indexes matching=[0, 3], sorted increasingly with duplicates forbidden. A matching has pairwise disjoint endpoints; the empty matching is allowed. Keep matching validity, maximality, maximum cardinality and perfect coverage distinct. Give (matching_indexes,matching_edges,conflicting_vertices,valid_matching?). List conflicting vertices in first-seen endpoint order while traversing the selected edges, not an invented numerical sort.

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

Read every selected original edge and count endpoint incidences. Any count above one is a conflict. Validity alone supplies no maximum-cardinality or perfect-matching certificate. The exact result, retaining the requested order, is ([0, 3], [[0, 1], [2, 3]], [], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r05-graph-matching-0013/)

Graph matching · qf-mod32-matching-audit · candidate





## 14. Selected Edges Form a Matching Only When Endpoints Are Disjoint

Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [1, 2], [1, 3], [2, 3]], endpoints increasing and the edge list lexicographically sorted. The source rejects noncanonical or duplicate edges. Select original edge indexes matching=[0, 4], sorted increasingly with duplicates forbidden. A matching has pairwise disjoint endpoints; the empty matching is allowed. Keep matching validity, maximality, maximum cardinality and perfect coverage distinct. Give (matching_indexes,matching_edges,conflicting_vertices,valid_matching?). List conflicting vertices in first-seen endpoint order while traversing the selected edges, not an invented numerical sort.

View answer and explanation([0, 4], [[0, 1], [2, 3]], [], Yes)

Read every selected original edge and count endpoint incidences. Any count above one is a conflict. Validity alone supplies no maximum-cardinality or perfect-matching certificate. The exact result, retaining the requested order, is ([0, 4], [[0, 1], [2, 3]], [], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r05-graph-matching-0014/)

Graph matching · qf-mod32-matching-audit · candidate





## 15. Selected Edges Form a Matching Only When Endpoints Are Disjoint

Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 3], [1, 2], [1, 3], [2, 3]], endpoints increasing and the edge list lexicographically sorted. The source rejects noncanonical or duplicate edges. Select original edge indexes matching=[0, 4], sorted increasingly with duplicates forbidden. A matching has pairwise disjoint endpoints; the empty matching is allowed. Keep matching validity, maximality, maximum cardinality and perfect coverage distinct. Give (matching_indexes,matching_edges,conflicting_vertices,valid_matching?). List conflicting vertices in first-seen endpoint order while traversing the selected edges, not an invented numerical sort.

View answer and explanation([0, 4], [[0, 1], [2, 3]], [], Yes)

Read every selected original edge and count endpoint incidences. Any count above one is a conflict. Validity alone supplies no maximum-cardinality or perfect-matching certificate. The exact result, retaining the requested order, is ([0, 4], [[0, 1], [2, 3]], [], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r05-graph-matching-0015/)

Graph matching · qf-mod32-matching-audit · candidate





## 16. Selected Edges Form a Matching Only When Endpoints Are Disjoint

Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 3], [1, 2], [1, 3], [2, 3]], endpoints increasing and the edge list lexicographically sorted. The source rejects noncanonical or duplicate edges. Select original edge indexes matching=[0, 5], sorted increasingly with duplicates forbidden. A matching has pairwise disjoint endpoints; the empty matching is allowed. Keep matching validity, maximality, maximum cardinality and perfect coverage distinct. Give (matching_indexes,matching_edges,conflicting_vertices,valid_matching?). List conflicting vertices in first-seen endpoint order while traversing the selected edges, not an invented numerical sort.

View answer and explanation([0, 5], [[0, 1], [2, 3]], [], Yes)

Read every selected original edge and count endpoint incidences. Any count above one is a conflict. Validity alone supplies no maximum-cardinality or perfect-matching certificate. The exact result, retaining the requested order, is ([0, 5], [[0, 1], [2, 3]], [], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r05-graph-matching-0016/)

Graph matching · qf-mod32-matching-audit · candidate





## 17. Selected Edges Form a Matching Only When Endpoints Are Disjoint

Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [2, 3], [4, 5]], endpoints increasing and the edge list lexicographically sorted. The source rejects noncanonical or duplicate edges. Select original edge indexes matching=[0, 1], sorted increasingly with duplicates forbidden. A matching has pairwise disjoint endpoints; the empty matching is allowed. Keep matching validity, maximality, maximum cardinality and perfect coverage distinct. Give (matching_indexes,matching_edges,conflicting_vertices,valid_matching?). List conflicting vertices in first-seen endpoint order while traversing the selected edges, not an invented numerical sort.

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

Read every selected original edge and count endpoint incidences. Any count above one is a conflict. Validity alone supplies no maximum-cardinality or perfect-matching certificate. The exact result, retaining the requested order, is ([0, 1], [[0, 1], [2, 3]], [], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r05-graph-matching-0017/)

Graph matching · qf-mod32-matching-audit · candidate





## 18. Selected Edges Form a Matching Only When Endpoints Are Disjoint

Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [2, 3], [4, 5]], endpoints increasing and the edge list lexicographically sorted. The source rejects noncanonical or duplicate edges. Select original edge indexes matching=[0, 2], sorted increasingly with duplicates forbidden. A matching has pairwise disjoint endpoints; the empty matching is allowed. Keep matching validity, maximality, maximum cardinality and perfect coverage distinct. Give (matching_indexes,matching_edges,conflicting_vertices,valid_matching?). List conflicting vertices in first-seen endpoint order while traversing the selected edges, not an invented numerical sort.

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

Read every selected original edge and count endpoint incidences. Any count above one is a conflict. Validity alone supplies no maximum-cardinality or perfect-matching certificate. The exact result, retaining the requested order, is ([0, 2], [[0, 1], [2, 3]], [], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r05-graph-matching-0018/)

Graph matching · qf-mod32-matching-audit · candidate





## 19. Selected Edges Form a Matching Only When Endpoints Are Disjoint

Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 3], [2, 3], [4, 5]], endpoints increasing and the edge list lexicographically sorted. The source rejects noncanonical or duplicate edges. Select original edge indexes matching=[0, 2], sorted increasingly with duplicates forbidden. A matching has pairwise disjoint endpoints; the empty matching is allowed. Keep matching validity, maximality, maximum cardinality and perfect coverage distinct. Give (matching_indexes,matching_edges,conflicting_vertices,valid_matching?). List conflicting vertices in first-seen endpoint order while traversing the selected edges, not an invented numerical sort.

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

Read every selected original edge and count endpoint incidences. Any count above one is a conflict. Validity alone supplies no maximum-cardinality or perfect-matching certificate. The exact result, retaining the requested order, is ([0, 2], [[0, 1], [2, 3]], [], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r05-graph-matching-0019/)

Graph matching · qf-mod32-matching-audit · candidate





## 20. Selected Edges Form a Matching Only When Endpoints Are Disjoint

Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 3], [2, 3], [4, 5]], endpoints increasing and the edge list lexicographically sorted. The source rejects noncanonical or duplicate edges. Select original edge indexes matching=[0, 3], sorted increasingly with duplicates forbidden. A matching has pairwise disjoint endpoints; the empty matching is allowed. Keep matching validity, maximality, maximum cardinality and perfect coverage distinct. Give (matching_indexes,matching_edges,conflicting_vertices,valid_matching?). List conflicting vertices in first-seen endpoint order while traversing the selected edges, not an invented numerical sort.

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

Read every selected original edge and count endpoint incidences. Any count above one is a conflict. Validity alone supplies no maximum-cardinality or perfect-matching certificate. The exact result, retaining the requested order, is ([0, 3], [[0, 1], [2, 3]], [], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r05-graph-matching-0020/)

Graph matching · qf-mod32-matching-audit · candidate





## 21. Selected Edges Form a Matching Only When Endpoints Are Disjoint

Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [1, 2], [2, 3], [4, 5]], endpoints increasing and the edge list lexicographically sorted. The source rejects noncanonical or duplicate edges. Select original edge indexes matching=[0, 2], sorted increasingly with duplicates forbidden. A matching has pairwise disjoint endpoints; the empty matching is allowed. Keep matching validity, maximality, maximum cardinality and perfect coverage distinct. Give (matching_indexes,matching_edges,conflicting_vertices,valid_matching?). List conflicting vertices in first-seen endpoint order while traversing the selected edges, not an invented numerical sort.

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

Read every selected original edge and count endpoint incidences. Any count above one is a conflict. Validity alone supplies no maximum-cardinality or perfect-matching certificate. The exact result, retaining the requested order, is ([0, 2], [[0, 1], [2, 3]], [], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r05-graph-matching-0021/)

Graph matching · qf-mod32-matching-audit · candidate





## 22. Selected Edges Form a Matching Only When Endpoints Are Disjoint

Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [1, 2], [2, 3], [4, 5]], endpoints increasing and the edge list lexicographically sorted. The source rejects noncanonical or duplicate edges. Select original edge indexes matching=[0, 3], sorted increasingly with duplicates forbidden. A matching has pairwise disjoint endpoints; the empty matching is allowed. Keep matching validity, maximality, maximum cardinality and perfect coverage distinct. Give (matching_indexes,matching_edges,conflicting_vertices,valid_matching?). List conflicting vertices in first-seen endpoint order while traversing the selected edges, not an invented numerical sort.

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

Read every selected original edge and count endpoint incidences. Any count above one is a conflict. Validity alone supplies no maximum-cardinality or perfect-matching certificate. The exact result, retaining the requested order, is ([0, 3], [[0, 1], [2, 3]], [], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r05-graph-matching-0022/)

Graph matching · qf-mod32-matching-audit · candidate





## 23. Selected Edges Form a Matching Only When Endpoints Are Disjoint

Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 3], [1, 2], [2, 3], [4, 5]], endpoints increasing and the edge list lexicographically sorted. The source rejects noncanonical or duplicate edges. Select original edge indexes matching=[0, 3], sorted increasingly with duplicates forbidden. A matching has pairwise disjoint endpoints; the empty matching is allowed. Keep matching validity, maximality, maximum cardinality and perfect coverage distinct. Give (matching_indexes,matching_edges,conflicting_vertices,valid_matching?). List conflicting vertices in first-seen endpoint order while traversing the selected edges, not an invented numerical sort.

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

Read every selected original edge and count endpoint incidences. Any count above one is a conflict. Validity alone supplies no maximum-cardinality or perfect-matching certificate. The exact result, retaining the requested order, is ([0, 3], [[0, 1], [2, 3]], [], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r05-graph-matching-0023/)

Graph matching · qf-mod32-matching-audit · candidate





## 24. Selected Edges Form a Matching Only When Endpoints Are Disjoint

Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 3], [1, 2], [2, 3], [4, 5]], endpoints increasing and the edge list lexicographically sorted. The source rejects noncanonical or duplicate edges. Select original edge indexes matching=[0, 4], sorted increasingly with duplicates forbidden. A matching has pairwise disjoint endpoints; the empty matching is allowed. Keep matching validity, maximality, maximum cardinality and perfect coverage distinct. Give (matching_indexes,matching_edges,conflicting_vertices,valid_matching?). List conflicting vertices in first-seen endpoint order while traversing the selected edges, not an invented numerical sort.

View answer and explanation([0, 4], [[0, 1], [2, 3]], [], Yes)

Read every selected original edge and count endpoint incidences. Any count above one is a conflict. Validity alone supplies no maximum-cardinality or perfect-matching certificate. The exact result, retaining the requested order, is ([0, 4], [[0, 1], [2, 3]], [], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r05-graph-matching-0024/)

Graph matching · qf-mod32-matching-audit · candidate





## 25. Selected Edges Form a Matching Only When Endpoints Are Disjoint

Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [1, 3], [2, 3], [4, 5]], endpoints increasing and the edge list lexicographically sorted. The source rejects noncanonical or duplicate edges. Select original edge indexes matching=[0, 2], sorted increasingly with duplicates forbidden. A matching has pairwise disjoint endpoints; the empty matching is allowed. Keep matching validity, maximality, maximum cardinality and perfect coverage distinct. Give (matching_indexes,matching_edges,conflicting_vertices,valid_matching?). List conflicting vertices in first-seen endpoint order while traversing the selected edges, not an invented numerical sort.

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

Read every selected original edge and count endpoint incidences. Any count above one is a conflict. Validity alone supplies no maximum-cardinality or perfect-matching certificate. The exact result, retaining the requested order, is ([0, 2], [[0, 1], [2, 3]], [], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r05-graph-matching-0025/)

Graph matching · qf-mod32-matching-audit · candidate





## 26. List the Whole Matched and Unmatched Vertex Sets

Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [2, 3]], endpoints increasing and the edge list lexicographically sorted. The source rejects noncanonical or duplicate edges. Select original edge indexes matching=[0, 1], sorted increasingly with duplicates forbidden. A matching has pairwise disjoint endpoints; the empty matching is allowed. Keep matching validity, maximality, maximum cardinality and perfect coverage distinct. Give (matching_edges,matched_vertices,unmatched_vertices,valid_matching?), with vertex lists increasing.

View answer and explanation([[0, 1], [2, 3]], [0, 1, 2, 3], [4, 5], Yes)

Take the union of selected edge endpoints and its complement in the complete carrier, retaining isolated vertices as unmatched where appropriate. Report validity separately even if an invalid selection still covers vertices. The exact result, retaining the requested order, is ([[0, 1], [2, 3]], [0, 1, 2, 3], [4, 5], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r05-graph-matching-0026/)

Graph matching · qf-mod32-matching-matched-vertices · candidate





## 27. List the Whole Matched and Unmatched Vertex Sets

Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [2, 3]], endpoints increasing and the edge list lexicographically sorted. The source rejects noncanonical or duplicate edges. Select original edge indexes matching=[0, 2], sorted increasingly with duplicates forbidden. A matching has pairwise disjoint endpoints; the empty matching is allowed. Keep matching validity, maximality, maximum cardinality and perfect coverage distinct. Give (matching_edges,matched_vertices,unmatched_vertices,valid_matching?), with vertex lists increasing.

View answer and explanation([[0, 1], [2, 3]], [0, 1, 2, 3], [4, 5], Yes)

Take the union of selected edge endpoints and its complement in the complete carrier, retaining isolated vertices as unmatched where appropriate. Report validity separately even if an invalid selection still covers vertices. The exact result, retaining the requested order, is ([[0, 1], [2, 3]], [0, 1, 2, 3], [4, 5], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r05-graph-matching-0027/)

Graph matching · qf-mod32-matching-matched-vertices · candidate





## 28. List the Whole Matched and Unmatched Vertex Sets

Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 3], [2, 3]], endpoints increasing and the edge list lexicographically sorted. The source rejects noncanonical or duplicate edges. Select original edge indexes matching=[0, 2], sorted increasingly with duplicates forbidden. A matching has pairwise disjoint endpoints; the empty matching is allowed. Keep matching validity, maximality, maximum cardinality and perfect coverage distinct. Give (matching_edges,matched_vertices,unmatched_vertices,valid_matching?), with vertex lists increasing.

View answer and explanation([[0, 1], [2, 3]], [0, 1, 2, 3], [4, 5], Yes)

Take the union of selected edge endpoints and its complement in the complete carrier, retaining isolated vertices as unmatched where appropriate. Report validity separately even if an invalid selection still covers vertices. The exact result, retaining the requested order, is ([[0, 1], [2, 3]], [0, 1, 2, 3], [4, 5], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r05-graph-matching-0028/)

Graph matching · qf-mod32-matching-matched-vertices · candidate





## 29. List the Whole Matched and Unmatched Vertex Sets

Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 3], [2, 3]], endpoints increasing and the edge list lexicographically sorted. The source rejects noncanonical or duplicate edges. Select original edge indexes matching=[0, 3], sorted increasingly with duplicates forbidden. A matching has pairwise disjoint endpoints; the empty matching is allowed. Keep matching validity, maximality, maximum cardinality and perfect coverage distinct. Give (matching_edges,matched_vertices,unmatched_vertices,valid_matching?), with vertex lists increasing.

View answer and explanation([[0, 1], [2, 3]], [0, 1, 2, 3], [4, 5], Yes)

Take the union of selected edge endpoints and its complement in the complete carrier, retaining isolated vertices as unmatched where appropriate. Report validity separately even if an invalid selection still covers vertices. The exact result, retaining the requested order, is ([[0, 1], [2, 3]], [0, 1, 2, 3], [4, 5], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r05-graph-matching-0029/)

Graph matching · qf-mod32-matching-matched-vertices · candidate





## 30. List the Whole Matched and Unmatched Vertex Sets

Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [1, 2], [2, 3]], endpoints increasing and the edge list lexicographically sorted. The source rejects noncanonical or duplicate edges. Select original edge indexes matching=[0, 2], sorted increasingly with duplicates forbidden. A matching has pairwise disjoint endpoints; the empty matching is allowed. Keep matching validity, maximality, maximum cardinality and perfect coverage distinct. Give (matching_edges,matched_vertices,unmatched_vertices,valid_matching?), with vertex lists increasing.

View answer and explanation([[0, 1], [2, 3]], [0, 1, 2, 3], [4, 5], Yes)

Take the union of selected edge endpoints and its complement in the complete carrier, retaining isolated vertices as unmatched where appropriate. Report validity separately even if an invalid selection still covers vertices. The exact result, retaining the requested order, is ([[0, 1], [2, 3]], [0, 1, 2, 3], [4, 5], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r05-graph-matching-0030/)

Graph matching · qf-mod32-matching-matched-vertices · candidate





## 31. List the Whole Matched and Unmatched Vertex Sets

Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [1, 2], [2, 3]], endpoints increasing and the edge list lexicographically sorted. The source rejects noncanonical or duplicate edges. Select original edge indexes matching=[0, 3], sorted increasingly with duplicates forbidden. A matching has pairwise disjoint endpoints; the empty matching is allowed. Keep matching validity, maximality, maximum cardinality and perfect coverage distinct. Give (matching_edges,matched_vertices,unmatched_vertices,valid_matching?), with vertex lists increasing.

View answer and explanation([[0, 1], [2, 3]], [0, 1, 2, 3], [4, 5], Yes)

Take the union of selected edge endpoints and its complement in the complete carrier, retaining isolated vertices as unmatched where appropriate. Report validity separately even if an invalid selection still covers vertices. The exact result, retaining the requested order, is ([[0, 1], [2, 3]], [0, 1, 2, 3], [4, 5], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r05-graph-matching-0031/)

Graph matching · qf-mod32-matching-matched-vertices · candidate





## 32. List the Whole Matched and Unmatched Vertex Sets

Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 3], [1, 2], [2, 3]], endpoints increasing and the edge list lexicographically sorted. The source rejects noncanonical or duplicate edges. Select original edge indexes matching=[0, 3], sorted increasingly with duplicates forbidden. A matching has pairwise disjoint endpoints; the empty matching is allowed. Keep matching validity, maximality, maximum cardinality and perfect coverage distinct. Give (matching_edges,matched_vertices,unmatched_vertices,valid_matching?), with vertex lists increasing.

View answer and explanation([[0, 1], [2, 3]], [0, 1, 2, 3], [4, 5], Yes)

Take the union of selected edge endpoints and its complement in the complete carrier, retaining isolated vertices as unmatched where appropriate. Report validity separately even if an invalid selection still covers vertices. The exact result, retaining the requested order, is ([[0, 1], [2, 3]], [0, 1, 2, 3], [4, 5], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r05-graph-matching-0032/)

Graph matching · qf-mod32-matching-matched-vertices · candidate





## 33. List the Whole Matched and Unmatched Vertex Sets

Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 3], [1, 2], [2, 3]], endpoints increasing and the edge list lexicographically sorted. The source rejects noncanonical or duplicate edges. Select original edge indexes matching=[0, 4], sorted increasingly with duplicates forbidden. A matching has pairwise disjoint endpoints; the empty matching is allowed. Keep matching validity, maximality, maximum cardinality and perfect coverage distinct. Give (matching_edges,matched_vertices,unmatched_vertices,valid_matching?), with vertex lists increasing.

View answer and explanation([[0, 1], [2, 3]], [0, 1, 2, 3], [4, 5], Yes)

Take the union of selected edge endpoints and its complement in the complete carrier, retaining isolated vertices as unmatched where appropriate. Report validity separately even if an invalid selection still covers vertices. The exact result, retaining the requested order, is ([[0, 1], [2, 3]], [0, 1, 2, 3], [4, 5], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r05-graph-matching-0033/)

Graph matching · qf-mod32-matching-matched-vertices · candidate





## 34. List the Whole Matched and Unmatched Vertex Sets

Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [1, 3], [2, 3]], endpoints increasing and the edge list lexicographically sorted. The source rejects noncanonical or duplicate edges. Select original edge indexes matching=[0, 2], sorted increasingly with duplicates forbidden. A matching has pairwise disjoint endpoints; the empty matching is allowed. Keep matching validity, maximality, maximum cardinality and perfect coverage distinct. Give (matching_edges,matched_vertices,unmatched_vertices,valid_matching?), with vertex lists increasing.

View answer and explanation([[0, 1], [2, 3]], [0, 1, 2, 3], [4, 5], Yes)

Take the union of selected edge endpoints and its complement in the complete carrier, retaining isolated vertices as unmatched where appropriate. Report validity separately even if an invalid selection still covers vertices. The exact result, retaining the requested order, is ([[0, 1], [2, 3]], [0, 1, 2, 3], [4, 5], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r05-graph-matching-0034/)

Graph matching · qf-mod32-matching-matched-vertices · candidate





## 35. List the Whole Matched and Unmatched Vertex Sets

Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [1, 3], [2, 3]], endpoints increasing and the edge list lexicographically sorted. The source rejects noncanonical or duplicate edges. Select original edge indexes matching=[0, 3], sorted increasingly with duplicates forbidden. A matching has pairwise disjoint endpoints; the empty matching is allowed. Keep matching validity, maximality, maximum cardinality and perfect coverage distinct. Give (matching_edges,matched_vertices,unmatched_vertices,valid_matching?), with vertex lists increasing.

View answer and explanation([[0, 1], [2, 3]], [0, 1, 2, 3], [4, 5], Yes)

Take the union of selected edge endpoints and its complement in the complete carrier, retaining isolated vertices as unmatched where appropriate. Report validity separately even if an invalid selection still covers vertices. The exact result, retaining the requested order, is ([[0, 1], [2, 3]], [0, 1, 2, 3], [4, 5], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r05-graph-matching-0035/)

Graph matching · qf-mod32-matching-matched-vertices · candidate





## 36. List the Whole Matched and Unmatched Vertex Sets

Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 3], [1, 3], [2, 3]], endpoints increasing and the edge list lexicographically sorted. The source rejects noncanonical or duplicate edges. Select original edge indexes matching=[0, 3], sorted increasingly with duplicates forbidden. A matching has pairwise disjoint endpoints; the empty matching is allowed. Keep matching validity, maximality, maximum cardinality and perfect coverage distinct. Give (matching_edges,matched_vertices,unmatched_vertices,valid_matching?), with vertex lists increasing.

View answer and explanation([[0, 1], [2, 3]], [0, 1, 2, 3], [4, 5], Yes)

Take the union of selected edge endpoints and its complement in the complete carrier, retaining isolated vertices as unmatched where appropriate. Report validity separately even if an invalid selection still covers vertices. The exact result, retaining the requested order, is ([[0, 1], [2, 3]], [0, 1, 2, 3], [4, 5], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r05-graph-matching-0036/)

Graph matching · qf-mod32-matching-matched-vertices · candidate





## 37. List the Whole Matched and Unmatched Vertex Sets

Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 3], [1, 3], [2, 3]], endpoints increasing and the edge list lexicographically sorted. The source rejects noncanonical or duplicate edges. Select original edge indexes matching=[0, 4], sorted increasingly with duplicates forbidden. A matching has pairwise disjoint endpoints; the empty matching is allowed. Keep matching validity, maximality, maximum cardinality and perfect coverage distinct. Give (matching_edges,matched_vertices,unmatched_vertices,valid_matching?), with vertex lists increasing.

View answer and explanation([[0, 1], [2, 3]], [0, 1, 2, 3], [4, 5], Yes)

Take the union of selected edge endpoints and its complement in the complete carrier, retaining isolated vertices as unmatched where appropriate. Report validity separately even if an invalid selection still covers vertices. The exact result, retaining the requested order, is ([[0, 1], [2, 3]], [0, 1, 2, 3], [4, 5], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r05-graph-matching-0037/)

Graph matching · qf-mod32-matching-matched-vertices · candidate





## 38. List the Whole Matched and Unmatched Vertex Sets

Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [1, 2], [1, 3], [2, 3]], endpoints increasing and the edge list lexicographically sorted. The source rejects noncanonical or duplicate edges. Select original edge indexes matching=[0, 3], sorted increasingly with duplicates forbidden. A matching has pairwise disjoint endpoints; the empty matching is allowed. Keep matching validity, maximality, maximum cardinality and perfect coverage distinct. Give (matching_edges,matched_vertices,unmatched_vertices,valid_matching?), with vertex lists increasing.

View answer and explanation([[0, 1], [2, 3]], [0, 1, 2, 3], [4, 5], Yes)

Take the union of selected edge endpoints and its complement in the complete carrier, retaining isolated vertices as unmatched where appropriate. Report validity separately even if an invalid selection still covers vertices. The exact result, retaining the requested order, is ([[0, 1], [2, 3]], [0, 1, 2, 3], [4, 5], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r05-graph-matching-0038/)

Graph matching · qf-mod32-matching-matched-vertices · candidate





## 39. List the Whole Matched and Unmatched Vertex Sets

Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [1, 2], [1, 3], [2, 3]], endpoints increasing and the edge list lexicographically sorted. The source rejects noncanonical or duplicate edges. Select original edge indexes matching=[0, 4], sorted increasingly with duplicates forbidden. A matching has pairwise disjoint endpoints; the empty matching is allowed. Keep matching validity, maximality, maximum cardinality and perfect coverage distinct. Give (matching_edges,matched_vertices,unmatched_vertices,valid_matching?), with vertex lists increasing.

View answer and explanation([[0, 1], [2, 3]], [0, 1, 2, 3], [4, 5], Yes)

Take the union of selected edge endpoints and its complement in the complete carrier, retaining isolated vertices as unmatched where appropriate. Report validity separately even if an invalid selection still covers vertices. The exact result, retaining the requested order, is ([[0, 1], [2, 3]], [0, 1, 2, 3], [4, 5], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r05-graph-matching-0039/)

Graph matching · qf-mod32-matching-matched-vertices · candidate





## 40. List the Whole Matched and Unmatched Vertex Sets

Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 3], [1, 2], [1, 3], [2, 3]], endpoints increasing and the edge list lexicographically sorted. The source rejects noncanonical or duplicate edges. Select original edge indexes matching=[0, 4], sorted increasingly with duplicates forbidden. A matching has pairwise disjoint endpoints; the empty matching is allowed. Keep matching validity, maximality, maximum cardinality and perfect coverage distinct. Give (matching_edges,matched_vertices,unmatched_vertices,valid_matching?), with vertex lists increasing.

View answer and explanation([[0, 1], [2, 3]], [0, 1, 2, 3], [4, 5], Yes)

Take the union of selected edge endpoints and its complement in the complete carrier, retaining isolated vertices as unmatched where appropriate. Report validity separately even if an invalid selection still covers vertices. The exact result, retaining the requested order, is ([[0, 1], [2, 3]], [0, 1, 2, 3], [4, 5], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r05-graph-matching-0040/)

Graph matching · qf-mod32-matching-matched-vertices · candidate





## 41. List the Whole Matched and Unmatched Vertex Sets

Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 3], [1, 2], [1, 3], [2, 3]], endpoints increasing and the edge list lexicographically sorted. The source rejects noncanonical or duplicate edges. Select original edge indexes matching=[0, 5], sorted increasingly with duplicates forbidden. A matching has pairwise disjoint endpoints; the empty matching is allowed. Keep matching validity, maximality, maximum cardinality and perfect coverage distinct. Give (matching_edges,matched_vertices,unmatched_vertices,valid_matching?), with vertex lists increasing.

View answer and explanation([[0, 1], [2, 3]], [0, 1, 2, 3], [4, 5], Yes)

Take the union of selected edge endpoints and its complement in the complete carrier, retaining isolated vertices as unmatched where appropriate. Report validity separately even if an invalid selection still covers vertices. The exact result, retaining the requested order, is ([[0, 1], [2, 3]], [0, 1, 2, 3], [4, 5], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r05-graph-matching-0041/)

Graph matching · qf-mod32-matching-matched-vertices · candidate





## 42. List the Whole Matched and Unmatched Vertex Sets

Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [2, 3], [4, 5]], endpoints increasing and the edge list lexicographically sorted. The source rejects noncanonical or duplicate edges. Select original edge indexes matching=[0, 1], sorted increasingly with duplicates forbidden. A matching has pairwise disjoint endpoints; the empty matching is allowed. Keep matching validity, maximality, maximum cardinality and perfect coverage distinct. Give (matching_edges,matched_vertices,unmatched_vertices,valid_matching?), with vertex lists increasing.

View answer and explanation([[0, 1], [2, 3]], [0, 1, 2, 3], [4, 5], Yes)

Take the union of selected edge endpoints and its complement in the complete carrier, retaining isolated vertices as unmatched where appropriate. Report validity separately even if an invalid selection still covers vertices. The exact result, retaining the requested order, is ([[0, 1], [2, 3]], [0, 1, 2, 3], [4, 5], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r05-graph-matching-0042/)

Graph matching · qf-mod32-matching-matched-vertices · candidate





## 43. List the Whole Matched and Unmatched Vertex Sets

Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [2, 3], [4, 5]], endpoints increasing and the edge list lexicographically sorted. The source rejects noncanonical or duplicate edges. Select original edge indexes matching=[0, 2], sorted increasingly with duplicates forbidden. A matching has pairwise disjoint endpoints; the empty matching is allowed. Keep matching validity, maximality, maximum cardinality and perfect coverage distinct. Give (matching_edges,matched_vertices,unmatched_vertices,valid_matching?), with vertex lists increasing.

View answer and explanation([[0, 1], [2, 3]], [0, 1, 2, 3], [4, 5], Yes)

Take the union of selected edge endpoints and its complement in the complete carrier, retaining isolated vertices as unmatched where appropriate. Report validity separately even if an invalid selection still covers vertices. The exact result, retaining the requested order, is ([[0, 1], [2, 3]], [0, 1, 2, 3], [4, 5], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r05-graph-matching-0043/)

Graph matching · qf-mod32-matching-matched-vertices · candidate





## 44. List the Whole Matched and Unmatched Vertex Sets

Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 3], [2, 3], [4, 5]], endpoints increasing and the edge list lexicographically sorted. The source rejects noncanonical or duplicate edges. Select original edge indexes matching=[0, 2], sorted increasingly with duplicates forbidden. A matching has pairwise disjoint endpoints; the empty matching is allowed. Keep matching validity, maximality, maximum cardinality and perfect coverage distinct. Give (matching_edges,matched_vertices,unmatched_vertices,valid_matching?), with vertex lists increasing.

View answer and explanation([[0, 1], [2, 3]], [0, 1, 2, 3], [4, 5], Yes)

Take the union of selected edge endpoints and its complement in the complete carrier, retaining isolated vertices as unmatched where appropriate. Report validity separately even if an invalid selection still covers vertices. The exact result, retaining the requested order, is ([[0, 1], [2, 3]], [0, 1, 2, 3], [4, 5], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r05-graph-matching-0044/)

Graph matching · qf-mod32-matching-matched-vertices · candidate





## 45. List the Whole Matched and Unmatched Vertex Sets

Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 3], [2, 3], [4, 5]], endpoints increasing and the edge list lexicographically sorted. The source rejects noncanonical or duplicate edges. Select original edge indexes matching=[0, 3], sorted increasingly with duplicates forbidden. A matching has pairwise disjoint endpoints; the empty matching is allowed. Keep matching validity, maximality, maximum cardinality and perfect coverage distinct. Give (matching_edges,matched_vertices,unmatched_vertices,valid_matching?), with vertex lists increasing.

View answer and explanation([[0, 1], [2, 3]], [0, 1, 2, 3], [4, 5], Yes)

Take the union of selected edge endpoints and its complement in the complete carrier, retaining isolated vertices as unmatched where appropriate. Report validity separately even if an invalid selection still covers vertices. The exact result, retaining the requested order, is ([[0, 1], [2, 3]], [0, 1, 2, 3], [4, 5], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r05-graph-matching-0045/)

Graph matching · qf-mod32-matching-matched-vertices · candidate





## 46. List the Whole Matched and Unmatched Vertex Sets

Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [1, 2], [2, 3], [4, 5]], endpoints increasing and the edge list lexicographically sorted. The source rejects noncanonical or duplicate edges. Select original edge indexes matching=[0, 2], sorted increasingly with duplicates forbidden. A matching has pairwise disjoint endpoints; the empty matching is allowed. Keep matching validity, maximality, maximum cardinality and perfect coverage distinct. Give (matching_edges,matched_vertices,unmatched_vertices,valid_matching?), with vertex lists increasing.

View answer and explanation([[0, 1], [2, 3]], [0, 1, 2, 3], [4, 5], Yes)

Take the union of selected edge endpoints and its complement in the complete carrier, retaining isolated vertices as unmatched where appropriate. Report validity separately even if an invalid selection still covers vertices. The exact result, retaining the requested order, is ([[0, 1], [2, 3]], [0, 1, 2, 3], [4, 5], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r05-graph-matching-0046/)

Graph matching · qf-mod32-matching-matched-vertices · candidate





## 47. List the Whole Matched and Unmatched Vertex Sets

Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [1, 2], [2, 3], [4, 5]], endpoints increasing and the edge list lexicographically sorted. The source rejects noncanonical or duplicate edges. Select original edge indexes matching=[0, 3], sorted increasingly with duplicates forbidden. A matching has pairwise disjoint endpoints; the empty matching is allowed. Keep matching validity, maximality, maximum cardinality and perfect coverage distinct. Give (matching_edges,matched_vertices,unmatched_vertices,valid_matching?), with vertex lists increasing.

View answer and explanation([[0, 1], [2, 3]], [0, 1, 2, 3], [4, 5], Yes)

Take the union of selected edge endpoints and its complement in the complete carrier, retaining isolated vertices as unmatched where appropriate. Report validity separately even if an invalid selection still covers vertices. The exact result, retaining the requested order, is ([[0, 1], [2, 3]], [0, 1, 2, 3], [4, 5], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r05-graph-matching-0047/)

Graph matching · qf-mod32-matching-matched-vertices · candidate





## 48. List the Whole Matched and Unmatched Vertex Sets

Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 3], [1, 2], [2, 3], [4, 5]], endpoints increasing and the edge list lexicographically sorted. The source rejects noncanonical or duplicate edges. Select original edge indexes matching=[0, 3], sorted increasingly with duplicates forbidden. A matching has pairwise disjoint endpoints; the empty matching is allowed. Keep matching validity, maximality, maximum cardinality and perfect coverage distinct. Give (matching_edges,matched_vertices,unmatched_vertices,valid_matching?), with vertex lists increasing.

View answer and explanation([[0, 1], [2, 3]], [0, 1, 2, 3], [4, 5], Yes)

Take the union of selected edge endpoints and its complement in the complete carrier, retaining isolated vertices as unmatched where appropriate. Report validity separately even if an invalid selection still covers vertices. The exact result, retaining the requested order, is ([[0, 1], [2, 3]], [0, 1, 2, 3], [4, 5], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r05-graph-matching-0048/)

Graph matching · qf-mod32-matching-matched-vertices · candidate





## 49. List the Whole Matched and Unmatched Vertex Sets

Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 3], [1, 2], [2, 3], [4, 5]], endpoints increasing and the edge list lexicographically sorted. The source rejects noncanonical or duplicate edges. Select original edge indexes matching=[0, 4], sorted increasingly with duplicates forbidden. A matching has pairwise disjoint endpoints; the empty matching is allowed. Keep matching validity, maximality, maximum cardinality and perfect coverage distinct. Give (matching_edges,matched_vertices,unmatched_vertices,valid_matching?), with vertex lists increasing.

View answer and explanation([[0, 1], [2, 3]], [0, 1, 2, 3], [4, 5], Yes)

Take the union of selected edge endpoints and its complement in the complete carrier, retaining isolated vertices as unmatched where appropriate. Report validity separately even if an invalid selection still covers vertices. The exact result, retaining the requested order, is ([[0, 1], [2, 3]], [0, 1, 2, 3], [4, 5], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r05-graph-matching-0049/)

Graph matching · qf-mod32-matching-matched-vertices · candidate





## 50. List the Whole Matched and Unmatched Vertex Sets

Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [1, 3], [2, 3], [4, 5]], endpoints increasing and the edge list lexicographically sorted. The source rejects noncanonical or duplicate edges. Select original edge indexes matching=[0, 2], sorted increasingly with duplicates forbidden. A matching has pairwise disjoint endpoints; the empty matching is allowed. Keep matching validity, maximality, maximum cardinality and perfect coverage distinct. Give (matching_edges,matched_vertices,unmatched_vertices,valid_matching?), with vertex lists increasing.

View answer and explanation([[0, 1], [2, 3]], [0, 1, 2, 3], [4, 5], Yes)

Take the union of selected edge endpoints and its complement in the complete carrier, retaining isolated vertices as unmatched where appropriate. Report validity separately even if an invalid selection still covers vertices. The exact result, retaining the requested order, is ([[0, 1], [2, 3]], [0, 1, 2, 3], [4, 5], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r05-graph-matching-0050/)

Graph matching · qf-mod32-matching-matched-vertices · candidate





## 51. A Maximal Matching May Still Have Smaller Cardinality than Another Matching

Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [2, 3]], endpoints increasing and the edge list lexicographically sorted. The source rejects noncanonical or duplicate edges. Select original edge indexes matching=[0, 1], sorted increasingly with duplicates forbidden. A matching has pairwise disjoint endpoints; the empty matching is allowed. Keep matching validity, maximality, maximum cardinality and perfect coverage distinct. Give (valid_matching?,addable_edge_indexes,maximal?,maximal_not_necessarily_maximum). List unselected original edges with both endpoints unmatched, in original index order.

View answer and explanation(Yes, [], Yes, maximal_not_necessarily_maximum)

A valid matching is maximal when no edge can be added directly. This does not rule out replacing some edges by a larger matching; maximum cardinality needs another certificate. The exact result, retaining the requested order, is (Yes, [], Yes, maximal_not_necessarily_maximum).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r05-graph-matching-0051/)

Graph matching · qf-mod32-matching-maximality-audit · candidate





## 52. A Maximal Matching May Still Have Smaller Cardinality than Another Matching

Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [2, 3]], endpoints increasing and the edge list lexicographically sorted. The source rejects noncanonical or duplicate edges. Select original edge indexes matching=[0, 2], sorted increasingly with duplicates forbidden. A matching has pairwise disjoint endpoints; the empty matching is allowed. Keep matching validity, maximality, maximum cardinality and perfect coverage distinct. Give (valid_matching?,addable_edge_indexes,maximal?,maximal_not_necessarily_maximum). List unselected original edges with both endpoints unmatched, in original index order.

View answer and explanation(Yes, [], Yes, maximal_not_necessarily_maximum)

A valid matching is maximal when no edge can be added directly. This does not rule out replacing some edges by a larger matching; maximum cardinality needs another certificate. The exact result, retaining the requested order, is (Yes, [], Yes, maximal_not_necessarily_maximum).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r05-graph-matching-0052/)

Graph matching · qf-mod32-matching-maximality-audit · candidate





## 53. A Maximal Matching May Still Have Smaller Cardinality than Another Matching

Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 3], [2, 3]], endpoints increasing and the edge list lexicographically sorted. The source rejects noncanonical or duplicate edges. Select original edge indexes matching=[0, 2], sorted increasingly with duplicates forbidden. A matching has pairwise disjoint endpoints; the empty matching is allowed. Keep matching validity, maximality, maximum cardinality and perfect coverage distinct. Give (valid_matching?,addable_edge_indexes,maximal?,maximal_not_necessarily_maximum). List unselected original edges with both endpoints unmatched, in original index order.

View answer and explanation(Yes, [], Yes, maximal_not_necessarily_maximum)

A valid matching is maximal when no edge can be added directly. This does not rule out replacing some edges by a larger matching; maximum cardinality needs another certificate. The exact result, retaining the requested order, is (Yes, [], Yes, maximal_not_necessarily_maximum).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r05-graph-matching-0053/)

Graph matching · qf-mod32-matching-maximality-audit · candidate





## 54. A Maximal Matching May Still Have Smaller Cardinality than Another Matching

Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 3], [2, 3]], endpoints increasing and the edge list lexicographically sorted. The source rejects noncanonical or duplicate edges. Select original edge indexes matching=[0, 3], sorted increasingly with duplicates forbidden. A matching has pairwise disjoint endpoints; the empty matching is allowed. Keep matching validity, maximality, maximum cardinality and perfect coverage distinct. Give (valid_matching?,addable_edge_indexes,maximal?,maximal_not_necessarily_maximum). List unselected original edges with both endpoints unmatched, in original index order.

View answer and explanation(Yes, [], Yes, maximal_not_necessarily_maximum)

A valid matching is maximal when no edge can be added directly. This does not rule out replacing some edges by a larger matching; maximum cardinality needs another certificate. The exact result, retaining the requested order, is (Yes, [], Yes, maximal_not_necessarily_maximum).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r05-graph-matching-0054/)

Graph matching · qf-mod32-matching-maximality-audit · candidate





## 55. A Maximal Matching May Still Have Smaller Cardinality than Another Matching

Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [1, 2], [2, 3]], endpoints increasing and the edge list lexicographically sorted. The source rejects noncanonical or duplicate edges. Select original edge indexes matching=[0, 2], sorted increasingly with duplicates forbidden. A matching has pairwise disjoint endpoints; the empty matching is allowed. Keep matching validity, maximality, maximum cardinality and perfect coverage distinct. Give (valid_matching?,addable_edge_indexes,maximal?,maximal_not_necessarily_maximum). List unselected original edges with both endpoints unmatched, in original index order.

View answer and explanation(Yes, [], Yes, maximal_not_necessarily_maximum)

A valid matching is maximal when no edge can be added directly. This does not rule out replacing some edges by a larger matching; maximum cardinality needs another certificate. The exact result, retaining the requested order, is (Yes, [], Yes, maximal_not_necessarily_maximum).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r05-graph-matching-0055/)

Graph matching · qf-mod32-matching-maximality-audit · candidate





## 56. A Maximal Matching May Still Have Smaller Cardinality than Another Matching

Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [1, 2], [2, 3]], endpoints increasing and the edge list lexicographically sorted. The source rejects noncanonical or duplicate edges. Select original edge indexes matching=[0, 3], sorted increasingly with duplicates forbidden. A matching has pairwise disjoint endpoints; the empty matching is allowed. Keep matching validity, maximality, maximum cardinality and perfect coverage distinct. Give (valid_matching?,addable_edge_indexes,maximal?,maximal_not_necessarily_maximum). List unselected original edges with both endpoints unmatched, in original index order.

View answer and explanation(Yes, [], Yes, maximal_not_necessarily_maximum)

A valid matching is maximal when no edge can be added directly. This does not rule out replacing some edges by a larger matching; maximum cardinality needs another certificate. The exact result, retaining the requested order, is (Yes, [], Yes, maximal_not_necessarily_maximum).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r05-graph-matching-0056/)

Graph matching · qf-mod32-matching-maximality-audit · candidate





## 57. A Maximal Matching May Still Have Smaller Cardinality than Another Matching

Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 3], [1, 2], [2, 3]], endpoints increasing and the edge list lexicographically sorted. The source rejects noncanonical or duplicate edges. Select original edge indexes matching=[0, 3], sorted increasingly with duplicates forbidden. A matching has pairwise disjoint endpoints; the empty matching is allowed. Keep matching validity, maximality, maximum cardinality and perfect coverage distinct. Give (valid_matching?,addable_edge_indexes,maximal?,maximal_not_necessarily_maximum). List unselected original edges with both endpoints unmatched, in original index order.

View answer and explanation(Yes, [], Yes, maximal_not_necessarily_maximum)

A valid matching is maximal when no edge can be added directly. This does not rule out replacing some edges by a larger matching; maximum cardinality needs another certificate. The exact result, retaining the requested order, is (Yes, [], Yes, maximal_not_necessarily_maximum).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r05-graph-matching-0057/)

Graph matching · qf-mod32-matching-maximality-audit · candidate





## 58. A Maximal Matching May Still Have Smaller Cardinality than Another Matching

Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 3], [1, 2], [2, 3]], endpoints increasing and the edge list lexicographically sorted. The source rejects noncanonical or duplicate edges. Select original edge indexes matching=[0, 4], sorted increasingly with duplicates forbidden. A matching has pairwise disjoint endpoints; the empty matching is allowed. Keep matching validity, maximality, maximum cardinality and perfect coverage distinct. Give (valid_matching?,addable_edge_indexes,maximal?,maximal_not_necessarily_maximum). List unselected original edges with both endpoints unmatched, in original index order.

View answer and explanation(Yes, [], Yes, maximal_not_necessarily_maximum)

A valid matching is maximal when no edge can be added directly. This does not rule out replacing some edges by a larger matching; maximum cardinality needs another certificate. The exact result, retaining the requested order, is (Yes, [], Yes, maximal_not_necessarily_maximum).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r05-graph-matching-0058/)

Graph matching · qf-mod32-matching-maximality-audit · candidate





## 59. A Maximal Matching May Still Have Smaller Cardinality than Another Matching

Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [1, 3], [2, 3]], endpoints increasing and the edge list lexicographically sorted. The source rejects noncanonical or duplicate edges. Select original edge indexes matching=[0, 2], sorted increasingly with duplicates forbidden. A matching has pairwise disjoint endpoints; the empty matching is allowed. Keep matching validity, maximality, maximum cardinality and perfect coverage distinct. Give (valid_matching?,addable_edge_indexes,maximal?,maximal_not_necessarily_maximum). List unselected original edges with both endpoints unmatched, in original index order.

View answer and explanation(Yes, [], Yes, maximal_not_necessarily_maximum)

A valid matching is maximal when no edge can be added directly. This does not rule out replacing some edges by a larger matching; maximum cardinality needs another certificate. The exact result, retaining the requested order, is (Yes, [], Yes, maximal_not_necessarily_maximum).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r05-graph-matching-0059/)

Graph matching · qf-mod32-matching-maximality-audit · candidate





## 60. A Maximal Matching May Still Have Smaller Cardinality than Another Matching

Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [1, 3], [2, 3]], endpoints increasing and the edge list lexicographically sorted. The source rejects noncanonical or duplicate edges. Select original edge indexes matching=[0, 3], sorted increasingly with duplicates forbidden. A matching has pairwise disjoint endpoints; the empty matching is allowed. Keep matching validity, maximality, maximum cardinality and perfect coverage distinct. Give (valid_matching?,addable_edge_indexes,maximal?,maximal_not_necessarily_maximum). List unselected original edges with both endpoints unmatched, in original index order.

View answer and explanation(Yes, [], Yes, maximal_not_necessarily_maximum)

A valid matching is maximal when no edge can be added directly. This does not rule out replacing some edges by a larger matching; maximum cardinality needs another certificate. The exact result, retaining the requested order, is (Yes, [], Yes, maximal_not_necessarily_maximum).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r05-graph-matching-0060/)

Graph matching · qf-mod32-matching-maximality-audit · candidate





## 61. A Maximal Matching May Still Have Smaller Cardinality than Another Matching

Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 3], [1, 3], [2, 3]], endpoints increasing and the edge list lexicographically sorted. The source rejects noncanonical or duplicate edges. Select original edge indexes matching=[0, 3], sorted increasingly with duplicates forbidden. A matching has pairwise disjoint endpoints; the empty matching is allowed. Keep matching validity, maximality, maximum cardinality and perfect coverage distinct. Give (valid_matching?,addable_edge_indexes,maximal?,maximal_not_necessarily_maximum). List unselected original edges with both endpoints unmatched, in original index order.

View answer and explanation(Yes, [], Yes, maximal_not_necessarily_maximum)

A valid matching is maximal when no edge can be added directly. This does not rule out replacing some edges by a larger matching; maximum cardinality needs another certificate. The exact result, retaining the requested order, is (Yes, [], Yes, maximal_not_necessarily_maximum).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r05-graph-matching-0061/)

Graph matching · qf-mod32-matching-maximality-audit · candidate





## 62. A Maximal Matching May Still Have Smaller Cardinality than Another Matching

Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 3], [1, 3], [2, 3]], endpoints increasing and the edge list lexicographically sorted. The source rejects noncanonical or duplicate edges. Select original edge indexes matching=[0, 4], sorted increasingly with duplicates forbidden. A matching has pairwise disjoint endpoints; the empty matching is allowed. Keep matching validity, maximality, maximum cardinality and perfect coverage distinct. Give (valid_matching?,addable_edge_indexes,maximal?,maximal_not_necessarily_maximum). List unselected original edges with both endpoints unmatched, in original index order.

View answer and explanation(Yes, [], Yes, maximal_not_necessarily_maximum)

A valid matching is maximal when no edge can be added directly. This does not rule out replacing some edges by a larger matching; maximum cardinality needs another certificate. The exact result, retaining the requested order, is (Yes, [], Yes, maximal_not_necessarily_maximum).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r05-graph-matching-0062/)

Graph matching · qf-mod32-matching-maximality-audit · candidate





## 63. A Maximal Matching May Still Have Smaller Cardinality than Another Matching

Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [1, 2], [1, 3], [2, 3]], endpoints increasing and the edge list lexicographically sorted. The source rejects noncanonical or duplicate edges. Select original edge indexes matching=[0, 3], sorted increasingly with duplicates forbidden. A matching has pairwise disjoint endpoints; the empty matching is allowed. Keep matching validity, maximality, maximum cardinality and perfect coverage distinct. Give (valid_matching?,addable_edge_indexes,maximal?,maximal_not_necessarily_maximum). List unselected original edges with both endpoints unmatched, in original index order.

View answer and explanation(Yes, [], Yes, maximal_not_necessarily_maximum)

A valid matching is maximal when no edge can be added directly. This does not rule out replacing some edges by a larger matching; maximum cardinality needs another certificate. The exact result, retaining the requested order, is (Yes, [], Yes, maximal_not_necessarily_maximum).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r05-graph-matching-0063/)

Graph matching · qf-mod32-matching-maximality-audit · candidate





## 64. A Maximal Matching May Still Have Smaller Cardinality than Another Matching

Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [1, 2], [1, 3], [2, 3]], endpoints increasing and the edge list lexicographically sorted. The source rejects noncanonical or duplicate edges. Select original edge indexes matching=[0, 4], sorted increasingly with duplicates forbidden. A matching has pairwise disjoint endpoints; the empty matching is allowed. Keep matching validity, maximality, maximum cardinality and perfect coverage distinct. Give (valid_matching?,addable_edge_indexes,maximal?,maximal_not_necessarily_maximum). List unselected original edges with both endpoints unmatched, in original index order.

View answer and explanation(Yes, [], Yes, maximal_not_necessarily_maximum)

A valid matching is maximal when no edge can be added directly. This does not rule out replacing some edges by a larger matching; maximum cardinality needs another certificate. The exact result, retaining the requested order, is (Yes, [], Yes, maximal_not_necessarily_maximum).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r05-graph-matching-0064/)

Graph matching · qf-mod32-matching-maximality-audit · candidate





## 65. A Maximal Matching May Still Have Smaller Cardinality than Another Matching

Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 3], [1, 2], [1, 3], [2, 3]], endpoints increasing and the edge list lexicographically sorted. The source rejects noncanonical or duplicate edges. Select original edge indexes matching=[0, 4], sorted increasingly with duplicates forbidden. A matching has pairwise disjoint endpoints; the empty matching is allowed. Keep matching validity, maximality, maximum cardinality and perfect coverage distinct. Give (valid_matching?,addable_edge_indexes,maximal?,maximal_not_necessarily_maximum). List unselected original edges with both endpoints unmatched, in original index order.

View answer and explanation(Yes, [], Yes, maximal_not_necessarily_maximum)

A valid matching is maximal when no edge can be added directly. This does not rule out replacing some edges by a larger matching; maximum cardinality needs another certificate. The exact result, retaining the requested order, is (Yes, [], Yes, maximal_not_necessarily_maximum).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r05-graph-matching-0065/)

Graph matching · qf-mod32-matching-maximality-audit · candidate





## 66. A Maximal Matching May Still Have Smaller Cardinality than Another Matching

Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 3], [1, 2], [1, 3], [2, 3]], endpoints increasing and the edge list lexicographically sorted. The source rejects noncanonical or duplicate edges. Select original edge indexes matching=[0, 5], sorted increasingly with duplicates forbidden. A matching has pairwise disjoint endpoints; the empty matching is allowed. Keep matching validity, maximality, maximum cardinality and perfect coverage distinct. Give (valid_matching?,addable_edge_indexes,maximal?,maximal_not_necessarily_maximum). List unselected original edges with both endpoints unmatched, in original index order.

View answer and explanation(Yes, [], Yes, maximal_not_necessarily_maximum)

A valid matching is maximal when no edge can be added directly. This does not rule out replacing some edges by a larger matching; maximum cardinality needs another certificate. The exact result, retaining the requested order, is (Yes, [], Yes, maximal_not_necessarily_maximum).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r05-graph-matching-0066/)

Graph matching · qf-mod32-matching-maximality-audit · candidate





## 67. A Maximal Matching May Still Have Smaller Cardinality than Another Matching

Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [2, 3], [4, 5]], endpoints increasing and the edge list lexicographically sorted. The source rejects noncanonical or duplicate edges. Select original edge indexes matching=[0, 1], sorted increasingly with duplicates forbidden. A matching has pairwise disjoint endpoints; the empty matching is allowed. Keep matching validity, maximality, maximum cardinality and perfect coverage distinct. Give (valid_matching?,addable_edge_indexes,maximal?,maximal_not_necessarily_maximum). List unselected original edges with both endpoints unmatched, in original index order.

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

A valid matching is maximal when no edge can be added directly. This does not rule out replacing some edges by a larger matching; maximum cardinality needs another certificate. The exact result, retaining the requested order, is (Yes, [2], No, maximal_not_necessarily_maximum).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r05-graph-matching-0067/)

Graph matching · qf-mod32-matching-maximality-audit · candidate





## 68. A Maximal Matching May Still Have Smaller Cardinality than Another Matching

Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [2, 3], [4, 5]], endpoints increasing and the edge list lexicographically sorted. The source rejects noncanonical or duplicate edges. Select original edge indexes matching=[0, 2], sorted increasingly with duplicates forbidden. A matching has pairwise disjoint endpoints; the empty matching is allowed. Keep matching validity, maximality, maximum cardinality and perfect coverage distinct. Give (valid_matching?,addable_edge_indexes,maximal?,maximal_not_necessarily_maximum). List unselected original edges with both endpoints unmatched, in original index order.

View answer and explanation(Yes, [3], No, maximal_not_necessarily_maximum)

A valid matching is maximal when no edge can be added directly. This does not rule out replacing some edges by a larger matching; maximum cardinality needs another certificate. The exact result, retaining the requested order, is (Yes, [3], No, maximal_not_necessarily_maximum).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r05-graph-matching-0068/)

Graph matching · qf-mod32-matching-maximality-audit · candidate





## 69. A Maximal Matching May Still Have Smaller Cardinality than Another Matching

Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 3], [2, 3], [4, 5]], endpoints increasing and the edge list lexicographically sorted. The source rejects noncanonical or duplicate edges. Select original edge indexes matching=[0, 2], sorted increasingly with duplicates forbidden. A matching has pairwise disjoint endpoints; the empty matching is allowed. Keep matching validity, maximality, maximum cardinality and perfect coverage distinct. Give (valid_matching?,addable_edge_indexes,maximal?,maximal_not_necessarily_maximum). List unselected original edges with both endpoints unmatched, in original index order.

View answer and explanation(Yes, [3], No, maximal_not_necessarily_maximum)

A valid matching is maximal when no edge can be added directly. This does not rule out replacing some edges by a larger matching; maximum cardinality needs another certificate. The exact result, retaining the requested order, is (Yes, [3], No, maximal_not_necessarily_maximum).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r05-graph-matching-0069/)

Graph matching · qf-mod32-matching-maximality-audit · candidate





## 70. A Maximal Matching May Still Have Smaller Cardinality than Another Matching

Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 3], [2, 3], [4, 5]], endpoints increasing and the edge list lexicographically sorted. The source rejects noncanonical or duplicate edges. Select original edge indexes matching=[0, 3], sorted increasingly with duplicates forbidden. A matching has pairwise disjoint endpoints; the empty matching is allowed. Keep matching validity, maximality, maximum cardinality and perfect coverage distinct. Give (valid_matching?,addable_edge_indexes,maximal?,maximal_not_necessarily_maximum). List unselected original edges with both endpoints unmatched, in original index order.

View answer and explanation(Yes, [4], No, maximal_not_necessarily_maximum)

A valid matching is maximal when no edge can be added directly. This does not rule out replacing some edges by a larger matching; maximum cardinality needs another certificate. The exact result, retaining the requested order, is (Yes, [4], No, maximal_not_necessarily_maximum).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r05-graph-matching-0070/)

Graph matching · qf-mod32-matching-maximality-audit · candidate





## 71. A Maximal Matching May Still Have Smaller Cardinality than Another Matching

Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [1, 2], [2, 3], [4, 5]], endpoints increasing and the edge list lexicographically sorted. The source rejects noncanonical or duplicate edges. Select original edge indexes matching=[0, 2], sorted increasingly with duplicates forbidden. A matching has pairwise disjoint endpoints; the empty matching is allowed. Keep matching validity, maximality, maximum cardinality and perfect coverage distinct. Give (valid_matching?,addable_edge_indexes,maximal?,maximal_not_necessarily_maximum). List unselected original edges with both endpoints unmatched, in original index order.

View answer and explanation(Yes, [3], No, maximal_not_necessarily_maximum)

A valid matching is maximal when no edge can be added directly. This does not rule out replacing some edges by a larger matching; maximum cardinality needs another certificate. The exact result, retaining the requested order, is (Yes, [3], No, maximal_not_necessarily_maximum).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r05-graph-matching-0071/)

Graph matching · qf-mod32-matching-maximality-audit · candidate





## 72. A Maximal Matching May Still Have Smaller Cardinality than Another Matching

Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [1, 2], [2, 3], [4, 5]], endpoints increasing and the edge list lexicographically sorted. The source rejects noncanonical or duplicate edges. Select original edge indexes matching=[0, 3], sorted increasingly with duplicates forbidden. A matching has pairwise disjoint endpoints; the empty matching is allowed. Keep matching validity, maximality, maximum cardinality and perfect coverage distinct. Give (valid_matching?,addable_edge_indexes,maximal?,maximal_not_necessarily_maximum). List unselected original edges with both endpoints unmatched, in original index order.

View answer and explanation(Yes, [4], No, maximal_not_necessarily_maximum)

A valid matching is maximal when no edge can be added directly. This does not rule out replacing some edges by a larger matching; maximum cardinality needs another certificate. The exact result, retaining the requested order, is (Yes, [4], No, maximal_not_necessarily_maximum).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r05-graph-matching-0072/)

Graph matching · qf-mod32-matching-maximality-audit · candidate





## 73. A Maximal Matching May Still Have Smaller Cardinality than Another Matching

Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 3], [1, 2], [2, 3], [4, 5]], endpoints increasing and the edge list lexicographically sorted. The source rejects noncanonical or duplicate edges. Select original edge indexes matching=[0, 3], sorted increasingly with duplicates forbidden. A matching has pairwise disjoint endpoints; the empty matching is allowed. Keep matching validity, maximality, maximum cardinality and perfect coverage distinct. Give (valid_matching?,addable_edge_indexes,maximal?,maximal_not_necessarily_maximum). List unselected original edges with both endpoints unmatched, in original index order.

View answer and explanation(Yes, [4], No, maximal_not_necessarily_maximum)

A valid matching is maximal when no edge can be added directly. This does not rule out replacing some edges by a larger matching; maximum cardinality needs another certificate. The exact result, retaining the requested order, is (Yes, [4], No, maximal_not_necessarily_maximum).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r05-graph-matching-0073/)

Graph matching · qf-mod32-matching-maximality-audit · candidate





## 74. A Maximal Matching May Still Have Smaller Cardinality than Another Matching

Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 3], [1, 2], [2, 3], [4, 5]], endpoints increasing and the edge list lexicographically sorted. The source rejects noncanonical or duplicate edges. Select original edge indexes matching=[0, 4], sorted increasingly with duplicates forbidden. A matching has pairwise disjoint endpoints; the empty matching is allowed. Keep matching validity, maximality, maximum cardinality and perfect coverage distinct. Give (valid_matching?,addable_edge_indexes,maximal?,maximal_not_necessarily_maximum). List unselected original edges with both endpoints unmatched, in original index order.

View answer and explanation(Yes, [5], No, maximal_not_necessarily_maximum)

A valid matching is maximal when no edge can be added directly. This does not rule out replacing some edges by a larger matching; maximum cardinality needs another certificate. The exact result, retaining the requested order, is (Yes, [5], No, maximal_not_necessarily_maximum).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r05-graph-matching-0074/)

Graph matching · qf-mod32-matching-maximality-audit · candidate





## 75. A Maximal Matching May Still Have Smaller Cardinality than Another Matching

Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [1, 3], [2, 3], [4, 5]], endpoints increasing and the edge list lexicographically sorted. The source rejects noncanonical or duplicate edges. Select original edge indexes matching=[0, 2], sorted increasingly with duplicates forbidden. A matching has pairwise disjoint endpoints; the empty matching is allowed. Keep matching validity, maximality, maximum cardinality and perfect coverage distinct. Give (valid_matching?,addable_edge_indexes,maximal?,maximal_not_necessarily_maximum). List unselected original edges with both endpoints unmatched, in original index order.

View answer and explanation(Yes, [3], No, maximal_not_necessarily_maximum)

A valid matching is maximal when no edge can be added directly. This does not rule out replacing some edges by a larger matching; maximum cardinality needs another certificate. The exact result, retaining the requested order, is (Yes, [3], No, maximal_not_necessarily_maximum).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r05-graph-matching-0075/)

Graph matching · qf-mod32-matching-maximality-audit · candidate





## 76. A Supplied Larger Valid Matching Refutes Maximum Cardinality

Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [2, 3]], endpoints increasing and the edge list lexicographically sorted. The source rejects noncanonical or duplicate edges. Select original edge indexes matching=[0], sorted increasingly with duplicates forbidden. A matching has pairwise disjoint endpoints; the empty matching is allowed. Keep matching validity, maximality, maximum cardinality and perfect coverage distinct. Compare larger_matching=[0, 1], also normalized as distinct original edge indexes. Give (matching_indexes,larger_indexes,size,larger_size,witness?,conclusion). The conclusion is supplied_matching_not_maximum or no_maximum_conclusion.

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

Validate both selections before comparing their sizes. A strictly larger valid witness disproves maximum cardinality of the first. No witness from this comparison does not prove that the first matching is maximum or perfect. The exact result, retaining the requested order, is ([0], [0, 1], 1, 2, Yes, supplied_matching_not_maximum).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r05-graph-matching-0076/)

Graph matching · qf-mod32-matching-maximum-distinction · candidate





## 77. A Supplied Larger Valid Matching Refutes Maximum Cardinality

Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [2, 3]], endpoints increasing and the edge list lexicographically sorted. The source rejects noncanonical or duplicate edges. Select original edge indexes matching=[0], sorted increasingly with duplicates forbidden. A matching has pairwise disjoint endpoints; the empty matching is allowed. Keep matching validity, maximality, maximum cardinality and perfect coverage distinct. Compare larger_matching=[0, 2], also normalized as distinct original edge indexes. Give (matching_indexes,larger_indexes,size,larger_size,witness?,conclusion). The conclusion is supplied_matching_not_maximum or no_maximum_conclusion.

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

Validate both selections before comparing their sizes. A strictly larger valid witness disproves maximum cardinality of the first. No witness from this comparison does not prove that the first matching is maximum or perfect. The exact result, retaining the requested order, is ([0], [0, 2], 1, 2, Yes, supplied_matching_not_maximum).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r05-graph-matching-0077/)

Graph matching · qf-mod32-matching-maximum-distinction · candidate





## 78. A Supplied Larger Valid Matching Refutes Maximum Cardinality

Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 3], [2, 3]], endpoints increasing and the edge list lexicographically sorted. The source rejects noncanonical or duplicate edges. Select original edge indexes matching=[0], sorted increasingly with duplicates forbidden. A matching has pairwise disjoint endpoints; the empty matching is allowed. Keep matching validity, maximality, maximum cardinality and perfect coverage distinct. Compare larger_matching=[0, 2], also normalized as distinct original edge indexes. Give (matching_indexes,larger_indexes,size,larger_size,witness?,conclusion). The conclusion is supplied_matching_not_maximum or no_maximum_conclusion.

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

Validate both selections before comparing their sizes. A strictly larger valid witness disproves maximum cardinality of the first. No witness from this comparison does not prove that the first matching is maximum or perfect. The exact result, retaining the requested order, is ([0], [0, 2], 1, 2, Yes, supplied_matching_not_maximum).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r05-graph-matching-0078/)

Graph matching · qf-mod32-matching-maximum-distinction · candidate





## 79. A Supplied Larger Valid Matching Refutes Maximum Cardinality

Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 3], [2, 3]], endpoints increasing and the edge list lexicographically sorted. The source rejects noncanonical or duplicate edges. Select original edge indexes matching=[0], sorted increasingly with duplicates forbidden. A matching has pairwise disjoint endpoints; the empty matching is allowed. Keep matching validity, maximality, maximum cardinality and perfect coverage distinct. Compare larger_matching=[0, 3], also normalized as distinct original edge indexes. Give (matching_indexes,larger_indexes,size,larger_size,witness?,conclusion). The conclusion is supplied_matching_not_maximum or no_maximum_conclusion.

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

Validate both selections before comparing their sizes. A strictly larger valid witness disproves maximum cardinality of the first. No witness from this comparison does not prove that the first matching is maximum or perfect. The exact result, retaining the requested order, is ([0], [0, 3], 1, 2, Yes, supplied_matching_not_maximum).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r05-graph-matching-0079/)

Graph matching · qf-mod32-matching-maximum-distinction · candidate





## 80. A Supplied Larger Valid Matching Refutes Maximum Cardinality

Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [1, 2], [2, 3]], endpoints increasing and the edge list lexicographically sorted. The source rejects noncanonical or duplicate edges. Select original edge indexes matching=[0], sorted increasingly with duplicates forbidden. A matching has pairwise disjoint endpoints; the empty matching is allowed. Keep matching validity, maximality, maximum cardinality and perfect coverage distinct. Compare larger_matching=[0, 2], also normalized as distinct original edge indexes. Give (matching_indexes,larger_indexes,size,larger_size,witness?,conclusion). The conclusion is supplied_matching_not_maximum or no_maximum_conclusion.

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

Validate both selections before comparing their sizes. A strictly larger valid witness disproves maximum cardinality of the first. No witness from this comparison does not prove that the first matching is maximum or perfect. The exact result, retaining the requested order, is ([0], [0, 2], 1, 2, Yes, supplied_matching_not_maximum).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r05-graph-matching-0080/)

Graph matching · qf-mod32-matching-maximum-distinction · candidate





## 81. A Supplied Larger Valid Matching Refutes Maximum Cardinality

Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [1, 2], [2, 3]], endpoints increasing and the edge list lexicographically sorted. The source rejects noncanonical or duplicate edges. Select original edge indexes matching=[0], sorted increasingly with duplicates forbidden. A matching has pairwise disjoint endpoints; the empty matching is allowed. Keep matching validity, maximality, maximum cardinality and perfect coverage distinct. Compare larger_matching=[0, 3], also normalized as distinct original edge indexes. Give (matching_indexes,larger_indexes,size,larger_size,witness?,conclusion). The conclusion is supplied_matching_not_maximum or no_maximum_conclusion.

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

Validate both selections before comparing their sizes. A strictly larger valid witness disproves maximum cardinality of the first. No witness from this comparison does not prove that the first matching is maximum or perfect. The exact result, retaining the requested order, is ([0], [0, 3], 1, 2, Yes, supplied_matching_not_maximum).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r05-graph-matching-0081/)

Graph matching · qf-mod32-matching-maximum-distinction · candidate





## 82. A Supplied Larger Valid Matching Refutes Maximum Cardinality

Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 3], [1, 2], [2, 3]], endpoints increasing and the edge list lexicographically sorted. The source rejects noncanonical or duplicate edges. Select original edge indexes matching=[0], sorted increasingly with duplicates forbidden. A matching has pairwise disjoint endpoints; the empty matching is allowed. Keep matching validity, maximality, maximum cardinality and perfect coverage distinct. Compare larger_matching=[0, 3], also normalized as distinct original edge indexes. Give (matching_indexes,larger_indexes,size,larger_size,witness?,conclusion). The conclusion is supplied_matching_not_maximum or no_maximum_conclusion.

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

Validate both selections before comparing their sizes. A strictly larger valid witness disproves maximum cardinality of the first. No witness from this comparison does not prove that the first matching is maximum or perfect. The exact result, retaining the requested order, is ([0], [0, 3], 1, 2, Yes, supplied_matching_not_maximum).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r05-graph-matching-0082/)

Graph matching · qf-mod32-matching-maximum-distinction · candidate





## 83. A Supplied Larger Valid Matching Refutes Maximum Cardinality

Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 3], [1, 2], [2, 3]], endpoints increasing and the edge list lexicographically sorted. The source rejects noncanonical or duplicate edges. Select original edge indexes matching=[0], sorted increasingly with duplicates forbidden. A matching has pairwise disjoint endpoints; the empty matching is allowed. Keep matching validity, maximality, maximum cardinality and perfect coverage distinct. Compare larger_matching=[0, 4], also normalized as distinct original edge indexes. Give (matching_indexes,larger_indexes,size,larger_size,witness?,conclusion). The conclusion is supplied_matching_not_maximum or no_maximum_conclusion.

View answer and explanation([0], [0, 4], 1, 2, Yes, supplied_matching_not_maximum)

Validate both selections before comparing their sizes. A strictly larger valid witness disproves maximum cardinality of the first. No witness from this comparison does not prove that the first matching is maximum or perfect. The exact result, retaining the requested order, is ([0], [0, 4], 1, 2, Yes, supplied_matching_not_maximum).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r05-graph-matching-0083/)

Graph matching · qf-mod32-matching-maximum-distinction · candidate





## 84. A Supplied Larger Valid Matching Refutes Maximum Cardinality

Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [1, 3], [2, 3]], endpoints increasing and the edge list lexicographically sorted. The source rejects noncanonical or duplicate edges. Select original edge indexes matching=[0], sorted increasingly with duplicates forbidden. A matching has pairwise disjoint endpoints; the empty matching is allowed. Keep matching validity, maximality, maximum cardinality and perfect coverage distinct. Compare larger_matching=[0, 2], also normalized as distinct original edge indexes. Give (matching_indexes,larger_indexes,size,larger_size,witness?,conclusion). The conclusion is supplied_matching_not_maximum or no_maximum_conclusion.

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

Validate both selections before comparing their sizes. A strictly larger valid witness disproves maximum cardinality of the first. No witness from this comparison does not prove that the first matching is maximum or perfect. The exact result, retaining the requested order, is ([0], [0, 2], 1, 2, Yes, supplied_matching_not_maximum).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r05-graph-matching-0084/)

Graph matching · qf-mod32-matching-maximum-distinction · candidate





## 85. A Supplied Larger Valid Matching Refutes Maximum Cardinality

Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [1, 3], [2, 3]], endpoints increasing and the edge list lexicographically sorted. The source rejects noncanonical or duplicate edges. Select original edge indexes matching=[0], sorted increasingly with duplicates forbidden. A matching has pairwise disjoint endpoints; the empty matching is allowed. Keep matching validity, maximality, maximum cardinality and perfect coverage distinct. Compare larger_matching=[1, 2], also normalized as distinct original edge indexes. Give (matching_indexes,larger_indexes,size,larger_size,witness?,conclusion). The conclusion is supplied_matching_not_maximum or no_maximum_conclusion.

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

Validate both selections before comparing their sizes. A strictly larger valid witness disproves maximum cardinality of the first. No witness from this comparison does not prove that the first matching is maximum or perfect. The exact result, retaining the requested order, is ([0], [1, 2], 1, 2, Yes, supplied_matching_not_maximum).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r05-graph-matching-0085/)

Graph matching · qf-mod32-matching-maximum-distinction · candidate





## 86. A Supplied Larger Valid Matching Refutes Maximum Cardinality

Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 3], [1, 3], [2, 3]], endpoints increasing and the edge list lexicographically sorted. The source rejects noncanonical or duplicate edges. Select original edge indexes matching=[0], sorted increasingly with duplicates forbidden. A matching has pairwise disjoint endpoints; the empty matching is allowed. Keep matching validity, maximality, maximum cardinality and perfect coverage distinct. Compare larger_matching=[0, 3], also normalized as distinct original edge indexes. Give (matching_indexes,larger_indexes,size,larger_size,witness?,conclusion). The conclusion is supplied_matching_not_maximum or no_maximum_conclusion.

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

Validate both selections before comparing their sizes. A strictly larger valid witness disproves maximum cardinality of the first. No witness from this comparison does not prove that the first matching is maximum or perfect. The exact result, retaining the requested order, is ([0], [0, 3], 1, 2, Yes, supplied_matching_not_maximum).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r05-graph-matching-0086/)

Graph matching · qf-mod32-matching-maximum-distinction · candidate





## 87. A Supplied Larger Valid Matching Refutes Maximum Cardinality

Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 3], [1, 3], [2, 3]], endpoints increasing and the edge list lexicographically sorted. The source rejects noncanonical or duplicate edges. Select original edge indexes matching=[0], sorted increasingly with duplicates forbidden. A matching has pairwise disjoint endpoints; the empty matching is allowed. Keep matching validity, maximality, maximum cardinality and perfect coverage distinct. Compare larger_matching=[1, 3], also normalized as distinct original edge indexes. Give (matching_indexes,larger_indexes,size,larger_size,witness?,conclusion). The conclusion is supplied_matching_not_maximum or no_maximum_conclusion.

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

Validate both selections before comparing their sizes. A strictly larger valid witness disproves maximum cardinality of the first. No witness from this comparison does not prove that the first matching is maximum or perfect. The exact result, retaining the requested order, is ([0], [1, 3], 1, 2, Yes, supplied_matching_not_maximum).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r05-graph-matching-0087/)

Graph matching · qf-mod32-matching-maximum-distinction · candidate





## 88. A Supplied Larger Valid Matching Refutes Maximum Cardinality

Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [1, 2], [1, 3], [2, 3]], endpoints increasing and the edge list lexicographically sorted. The source rejects noncanonical or duplicate edges. Select original edge indexes matching=[0], sorted increasingly with duplicates forbidden. A matching has pairwise disjoint endpoints; the empty matching is allowed. Keep matching validity, maximality, maximum cardinality and perfect coverage distinct. Compare larger_matching=[0, 3], also normalized as distinct original edge indexes. Give (matching_indexes,larger_indexes,size,larger_size,witness?,conclusion). The conclusion is supplied_matching_not_maximum or no_maximum_conclusion.

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

Validate both selections before comparing their sizes. A strictly larger valid witness disproves maximum cardinality of the first. No witness from this comparison does not prove that the first matching is maximum or perfect. The exact result, retaining the requested order, is ([0], [0, 3], 1, 2, Yes, supplied_matching_not_maximum).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r05-graph-matching-0088/)

Graph matching · qf-mod32-matching-maximum-distinction · candidate





## 89. A Supplied Larger Valid Matching Refutes Maximum Cardinality

Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [1, 2], [1, 3], [2, 3]], endpoints increasing and the edge list lexicographically sorted. The source rejects noncanonical or duplicate edges. Select original edge indexes matching=[0], sorted increasingly with duplicates forbidden. A matching has pairwise disjoint endpoints; the empty matching is allowed. Keep matching validity, maximality, maximum cardinality and perfect coverage distinct. Compare larger_matching=[1, 3], also normalized as distinct original edge indexes. Give (matching_indexes,larger_indexes,size,larger_size,witness?,conclusion). The conclusion is supplied_matching_not_maximum or no_maximum_conclusion.

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

Validate both selections before comparing their sizes. A strictly larger valid witness disproves maximum cardinality of the first. No witness from this comparison does not prove that the first matching is maximum or perfect. The exact result, retaining the requested order, is ([0], [1, 3], 1, 2, Yes, supplied_matching_not_maximum).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r05-graph-matching-0089/)

Graph matching · qf-mod32-matching-maximum-distinction · candidate





## 90. A Supplied Larger Valid Matching Refutes Maximum Cardinality

Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 3], [1, 2], [1, 3], [2, 3]], endpoints increasing and the edge list lexicographically sorted. The source rejects noncanonical or duplicate edges. Select original edge indexes matching=[0], sorted increasingly with duplicates forbidden. A matching has pairwise disjoint endpoints; the empty matching is allowed. Keep matching validity, maximality, maximum cardinality and perfect coverage distinct. Compare larger_matching=[0, 4], also normalized as distinct original edge indexes. Give (matching_indexes,larger_indexes,size,larger_size,witness?,conclusion). The conclusion is supplied_matching_not_maximum or no_maximum_conclusion.

View answer and explanation([0], [0, 4], 1, 2, Yes, supplied_matching_not_maximum)

Validate both selections before comparing their sizes. A strictly larger valid witness disproves maximum cardinality of the first. No witness from this comparison does not prove that the first matching is maximum or perfect. The exact result, retaining the requested order, is ([0], [0, 4], 1, 2, Yes, supplied_matching_not_maximum).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r05-graph-matching-0090/)

Graph matching · qf-mod32-matching-maximum-distinction · candidate





## 91. A Supplied Larger Valid Matching Refutes Maximum Cardinality

Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 3], [1, 2], [1, 3], [2, 3]], endpoints increasing and the edge list lexicographically sorted. The source rejects noncanonical or duplicate edges. Select original edge indexes matching=[0], sorted increasingly with duplicates forbidden. A matching has pairwise disjoint endpoints; the empty matching is allowed. Keep matching validity, maximality, maximum cardinality and perfect coverage distinct. Compare larger_matching=[1, 4], also normalized as distinct original edge indexes. Give (matching_indexes,larger_indexes,size,larger_size,witness?,conclusion). The conclusion is supplied_matching_not_maximum or no_maximum_conclusion.

View answer and explanation([0], [1, 4], 1, 2, Yes, supplied_matching_not_maximum)

Validate both selections before comparing their sizes. A strictly larger valid witness disproves maximum cardinality of the first. No witness from this comparison does not prove that the first matching is maximum or perfect. The exact result, retaining the requested order, is ([0], [1, 4], 1, 2, Yes, supplied_matching_not_maximum).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r05-graph-matching-0091/)

Graph matching · qf-mod32-matching-maximum-distinction · candidate





## 92. A Supplied Larger Valid Matching Refutes Maximum Cardinality

Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [2, 3], [4, 5]], endpoints increasing and the edge list lexicographically sorted. The source rejects noncanonical or duplicate edges. Select original edge indexes matching=[0], sorted increasingly with duplicates forbidden. A matching has pairwise disjoint endpoints; the empty matching is allowed. Keep matching validity, maximality, maximum cardinality and perfect coverage distinct. Compare larger_matching=[0, 1], also normalized as distinct original edge indexes. Give (matching_indexes,larger_indexes,size,larger_size,witness?,conclusion). The conclusion is supplied_matching_not_maximum or no_maximum_conclusion.

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

Validate both selections before comparing their sizes. A strictly larger valid witness disproves maximum cardinality of the first. No witness from this comparison does not prove that the first matching is maximum or perfect. The exact result, retaining the requested order, is ([0], [0, 1], 1, 2, Yes, supplied_matching_not_maximum).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r05-graph-matching-0092/)

Graph matching · qf-mod32-matching-maximum-distinction · candidate





## 93. A Supplied Larger Valid Matching Refutes Maximum Cardinality

Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [2, 3], [4, 5]], endpoints increasing and the edge list lexicographically sorted. The source rejects noncanonical or duplicate edges. Select original edge indexes matching=[0], sorted increasingly with duplicates forbidden. A matching has pairwise disjoint endpoints; the empty matching is allowed. Keep matching validity, maximality, maximum cardinality and perfect coverage distinct. Compare larger_matching=[0, 2], also normalized as distinct original edge indexes. Give (matching_indexes,larger_indexes,size,larger_size,witness?,conclusion). The conclusion is supplied_matching_not_maximum or no_maximum_conclusion.

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

Validate both selections before comparing their sizes. A strictly larger valid witness disproves maximum cardinality of the first. No witness from this comparison does not prove that the first matching is maximum or perfect. The exact result, retaining the requested order, is ([0], [0, 2], 1, 2, Yes, supplied_matching_not_maximum).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r05-graph-matching-0093/)

Graph matching · qf-mod32-matching-maximum-distinction · candidate





## 94. A Supplied Larger Valid Matching Refutes Maximum Cardinality

Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 3], [2, 3], [4, 5]], endpoints increasing and the edge list lexicographically sorted. The source rejects noncanonical or duplicate edges. Select original edge indexes matching=[0], sorted increasingly with duplicates forbidden. A matching has pairwise disjoint endpoints; the empty matching is allowed. Keep matching validity, maximality, maximum cardinality and perfect coverage distinct. Compare larger_matching=[0, 2], also normalized as distinct original edge indexes. Give (matching_indexes,larger_indexes,size,larger_size,witness?,conclusion). The conclusion is supplied_matching_not_maximum or no_maximum_conclusion.

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

Validate both selections before comparing their sizes. A strictly larger valid witness disproves maximum cardinality of the first. No witness from this comparison does not prove that the first matching is maximum or perfect. The exact result, retaining the requested order, is ([0], [0, 2], 1, 2, Yes, supplied_matching_not_maximum).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r05-graph-matching-0094/)

Graph matching · qf-mod32-matching-maximum-distinction · candidate





## 95. A Supplied Larger Valid Matching Refutes Maximum Cardinality

Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 3], [2, 3], [4, 5]], endpoints increasing and the edge list lexicographically sorted. The source rejects noncanonical or duplicate edges. Select original edge indexes matching=[0], sorted increasingly with duplicates forbidden. A matching has pairwise disjoint endpoints; the empty matching is allowed. Keep matching validity, maximality, maximum cardinality and perfect coverage distinct. Compare larger_matching=[0, 3], also normalized as distinct original edge indexes. Give (matching_indexes,larger_indexes,size,larger_size,witness?,conclusion). The conclusion is supplied_matching_not_maximum or no_maximum_conclusion.

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

Validate both selections before comparing their sizes. A strictly larger valid witness disproves maximum cardinality of the first. No witness from this comparison does not prove that the first matching is maximum or perfect. The exact result, retaining the requested order, is ([0], [0, 3], 1, 2, Yes, supplied_matching_not_maximum).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r05-graph-matching-0095/)

Graph matching · qf-mod32-matching-maximum-distinction · candidate





## 96. A Supplied Larger Valid Matching Refutes Maximum Cardinality

Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [1, 2], [2, 3], [4, 5]], endpoints increasing and the edge list lexicographically sorted. The source rejects noncanonical or duplicate edges. Select original edge indexes matching=[0], sorted increasingly with duplicates forbidden. A matching has pairwise disjoint endpoints; the empty matching is allowed. Keep matching validity, maximality, maximum cardinality and perfect coverage distinct. Compare larger_matching=[0, 2], also normalized as distinct original edge indexes. Give (matching_indexes,larger_indexes,size,larger_size,witness?,conclusion). The conclusion is supplied_matching_not_maximum or no_maximum_conclusion.

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

Validate both selections before comparing their sizes. A strictly larger valid witness disproves maximum cardinality of the first. No witness from this comparison does not prove that the first matching is maximum or perfect. The exact result, retaining the requested order, is ([0], [0, 2], 1, 2, Yes, supplied_matching_not_maximum).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r05-graph-matching-0096/)

Graph matching · qf-mod32-matching-maximum-distinction · candidate





## 97. A Supplied Larger Valid Matching Refutes Maximum Cardinality

Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [1, 2], [2, 3], [4, 5]], endpoints increasing and the edge list lexicographically sorted. The source rejects noncanonical or duplicate edges. Select original edge indexes matching=[0], sorted increasingly with duplicates forbidden. A matching has pairwise disjoint endpoints; the empty matching is allowed. Keep matching validity, maximality, maximum cardinality and perfect coverage distinct. Compare larger_matching=[0, 3], also normalized as distinct original edge indexes. Give (matching_indexes,larger_indexes,size,larger_size,witness?,conclusion). The conclusion is supplied_matching_not_maximum or no_maximum_conclusion.

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

Validate both selections before comparing their sizes. A strictly larger valid witness disproves maximum cardinality of the first. No witness from this comparison does not prove that the first matching is maximum or perfect. The exact result, retaining the requested order, is ([0], [0, 3], 1, 2, Yes, supplied_matching_not_maximum).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r05-graph-matching-0097/)

Graph matching · qf-mod32-matching-maximum-distinction · candidate





## 98. A Supplied Larger Valid Matching Refutes Maximum Cardinality

Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 3], [1, 2], [2, 3], [4, 5]], endpoints increasing and the edge list lexicographically sorted. The source rejects noncanonical or duplicate edges. Select original edge indexes matching=[0], sorted increasingly with duplicates forbidden. A matching has pairwise disjoint endpoints; the empty matching is allowed. Keep matching validity, maximality, maximum cardinality and perfect coverage distinct. Compare larger_matching=[0, 3], also normalized as distinct original edge indexes. Give (matching_indexes,larger_indexes,size,larger_size,witness?,conclusion). The conclusion is supplied_matching_not_maximum or no_maximum_conclusion.

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

Validate both selections before comparing their sizes. A strictly larger valid witness disproves maximum cardinality of the first. No witness from this comparison does not prove that the first matching is maximum or perfect. The exact result, retaining the requested order, is ([0], [0, 3], 1, 2, Yes, supplied_matching_not_maximum).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r05-graph-matching-0098/)

Graph matching · qf-mod32-matching-maximum-distinction · candidate





## 99. A Supplied Larger Valid Matching Refutes Maximum Cardinality

Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 3], [1, 2], [2, 3], [4, 5]], endpoints increasing and the edge list lexicographically sorted. The source rejects noncanonical or duplicate edges. Select original edge indexes matching=[0], sorted increasingly with duplicates forbidden. A matching has pairwise disjoint endpoints; the empty matching is allowed. Keep matching validity, maximality, maximum cardinality and perfect coverage distinct. Compare larger_matching=[0, 4], also normalized as distinct original edge indexes. Give (matching_indexes,larger_indexes,size,larger_size,witness?,conclusion). The conclusion is supplied_matching_not_maximum or no_maximum_conclusion.

View answer and explanation([0], [0, 4], 1, 2, Yes, supplied_matching_not_maximum)

Validate both selections before comparing their sizes. A strictly larger valid witness disproves maximum cardinality of the first. No witness from this comparison does not prove that the first matching is maximum or perfect. The exact result, retaining the requested order, is ([0], [0, 4], 1, 2, Yes, supplied_matching_not_maximum).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r05-graph-matching-0099/)

Graph matching · qf-mod32-matching-maximum-distinction · candidate





## 100. A Supplied Larger Valid Matching Refutes Maximum Cardinality

Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [1, 3], [2, 3], [4, 5]], endpoints increasing and the edge list lexicographically sorted. The source rejects noncanonical or duplicate edges. Select original edge indexes matching=[0], sorted increasingly with duplicates forbidden. A matching has pairwise disjoint endpoints; the empty matching is allowed. Keep matching validity, maximality, maximum cardinality and perfect coverage distinct. Compare larger_matching=[0, 2], also normalized as distinct original edge indexes. Give (matching_indexes,larger_indexes,size,larger_size,witness?,conclusion). The conclusion is supplied_matching_not_maximum or no_maximum_conclusion.

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

Validate both selections before comparing their sizes. A strictly larger valid witness disproves maximum cardinality of the first. No witness from this comparison does not prove that the first matching is maximum or perfect. The exact result, retaining the requested order, is ([0], [0, 2], 1, 2, Yes, supplied_matching_not_maximum).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r05-graph-matching-0100/)

Graph matching · qf-mod32-matching-maximum-distinction · candidate



[Canonical question JSONL](https://ocme.evemisslab.com/data/questions/batches/batch-20261014-advance-20261101-r05-graph-matching-foundations/questions.jsonl) · [English JSONL](https://ocme.evemisslab.com/data/i18n/questions/batch-20261014-advance-20261101-r05-graph-matching-foundations.en.jsonl) · [Validation report](https://ocme.evemisslab.com/data/questions/batches/batch-20261014-advance-20261101-r05-graph-matching-foundations/validation-report.json)