Maximum Matchings: Validity before Global Optimality
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batch-20261014-advance-20261103-r06-maximum-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. The Maximum Size Requires All Valid Matchings
Use the complete bounded simple graph on Fin(6)={0,…,5} with canonical ordered undirected edges=[[0, 1], [2, 3]]. Edge indices are zero-based positions in this exact list. A matching contains distinct edge indices with no shared endpoint. Enumerate every edge subset to determine the global maximum size; include the empty matching. List indices within each matching increasingly and enumerate matchings in increasing edge-subset mask order. Maximum,inclusion-maximal,perfect and unique are distinct properties. Give (matching_count,maximum_size,maximum_matchings,exhaustive_all_edge_subsets).
View answer and explanation
(4, 2, [[0, 1]], exhaustive_all_edge_subsets)
Reject every edge subset with a repeated endpoint,then maximize size over all remaining matchings. Count all valid matchings including the empty one and retain every maximum. This finite enumeration is not a general complexity or perfect-matching theorem. The exact result, retaining the requested order, is (4, 2, [[0, 1]], exhaustive_all_edge_subsets).
Reviewed step-by-step derivation
Maximum matching · qf-mod34-maxmatching-maximum-size · candidate
2. The Maximum Size Requires All Valid Matchings
Use the complete bounded simple graph on Fin(6)={0,…,5} with canonical ordered undirected edges=[[0, 1], [0, 2], [2, 3]]. Edge indices are zero-based positions in this exact list. A matching contains distinct edge indices with no shared endpoint. Enumerate every edge subset to determine the global maximum size; include the empty matching. List indices within each matching increasingly and enumerate matchings in increasing edge-subset mask order. Maximum,inclusion-maximal,perfect and unique are distinct properties. Give (matching_count,maximum_size,maximum_matchings,exhaustive_all_edge_subsets).
View answer and explanation
(5, 2, [[0, 2]], exhaustive_all_edge_subsets)
Reject every edge subset with a repeated endpoint,then maximize size over all remaining matchings. Count all valid matchings including the empty one and retain every maximum. This finite enumeration is not a general complexity or perfect-matching theorem. The exact result, retaining the requested order, is (5, 2, [[0, 2]], exhaustive_all_edge_subsets).
Reviewed step-by-step derivation
Maximum matching · qf-mod34-maxmatching-maximum-size · candidate
3. The Maximum Size Requires All Valid Matchings
Use the complete bounded simple graph on Fin(6)={0,…,5} with canonical ordered undirected edges=[[0, 1], [0, 3], [2, 3]]. Edge indices are zero-based positions in this exact list. A matching contains distinct edge indices with no shared endpoint. Enumerate every edge subset to determine the global maximum size; include the empty matching. List indices within each matching increasingly and enumerate matchings in increasing edge-subset mask order. Maximum,inclusion-maximal,perfect and unique are distinct properties. Give (matching_count,maximum_size,maximum_matchings,exhaustive_all_edge_subsets).
View answer and explanation
(5, 2, [[0, 2]], exhaustive_all_edge_subsets)
Reject every edge subset with a repeated endpoint,then maximize size over all remaining matchings. Count all valid matchings including the empty one and retain every maximum. This finite enumeration is not a general complexity or perfect-matching theorem. The exact result, retaining the requested order, is (5, 2, [[0, 2]], exhaustive_all_edge_subsets).
Reviewed step-by-step derivation
Maximum matching · qf-mod34-maxmatching-maximum-size · candidate
4. The Maximum Size Requires All Valid Matchings
Use the complete bounded simple graph on Fin(6)={0,…,5} with canonical ordered undirected edges=[[0, 1], [0, 2], [0, 3], [2, 3]]. Edge indices are zero-based positions in this exact list. A matching contains distinct edge indices with no shared endpoint. Enumerate every edge subset to determine the global maximum size; include the empty matching. List indices within each matching increasingly and enumerate matchings in increasing edge-subset mask order. Maximum,inclusion-maximal,perfect and unique are distinct properties. Give (matching_count,maximum_size,maximum_matchings,exhaustive_all_edge_subsets).
View answer and explanation
(6, 2, [[0, 3]], exhaustive_all_edge_subsets)
Reject every edge subset with a repeated endpoint,then maximize size over all remaining matchings. Count all valid matchings including the empty one and retain every maximum. This finite enumeration is not a general complexity or perfect-matching theorem. The exact result, retaining the requested order, is (6, 2, [[0, 3]], exhaustive_all_edge_subsets).
Reviewed step-by-step derivation
Maximum matching · qf-mod34-maxmatching-maximum-size · candidate
5. The Maximum Size Requires All Valid Matchings
Use the complete bounded simple graph on Fin(6)={0,…,5} with canonical ordered undirected edges=[[0, 1], [1, 2], [2, 3]]. Edge indices are zero-based positions in this exact list. A matching contains distinct edge indices with no shared endpoint. Enumerate every edge subset to determine the global maximum size; include the empty matching. List indices within each matching increasingly and enumerate matchings in increasing edge-subset mask order. Maximum,inclusion-maximal,perfect and unique are distinct properties. Give (matching_count,maximum_size,maximum_matchings,exhaustive_all_edge_subsets).
View answer and explanation
(5, 2, [[0, 2]], exhaustive_all_edge_subsets)
Reject every edge subset with a repeated endpoint,then maximize size over all remaining matchings. Count all valid matchings including the empty one and retain every maximum. This finite enumeration is not a general complexity or perfect-matching theorem. The exact result, retaining the requested order, is (5, 2, [[0, 2]], exhaustive_all_edge_subsets).
Reviewed step-by-step derivation
Maximum matching · qf-mod34-maxmatching-maximum-size · candidate
6. The Maximum Size Requires All Valid Matchings
Use the complete bounded simple graph on Fin(6)={0,…,5} with canonical ordered undirected edges=[[0, 1], [0, 2], [1, 2], [2, 3]]. Edge indices are zero-based positions in this exact list. A matching contains distinct edge indices with no shared endpoint. Enumerate every edge subset to determine the global maximum size; include the empty matching. List indices within each matching increasingly and enumerate matchings in increasing edge-subset mask order. Maximum,inclusion-maximal,perfect and unique are distinct properties. Give (matching_count,maximum_size,maximum_matchings,exhaustive_all_edge_subsets).
View answer and explanation
(6, 2, [[0, 3]], exhaustive_all_edge_subsets)
Reject every edge subset with a repeated endpoint,then maximize size over all remaining matchings. Count all valid matchings including the empty one and retain every maximum. This finite enumeration is not a general complexity or perfect-matching theorem. The exact result, retaining the requested order, is (6, 2, [[0, 3]], exhaustive_all_edge_subsets).
Reviewed step-by-step derivation
Maximum matching · qf-mod34-maxmatching-maximum-size · candidate
7. The Maximum Size Requires All Valid Matchings
Use the complete bounded simple graph on Fin(6)={0,…,5} with canonical ordered undirected edges=[[0, 1], [0, 3], [1, 2], [2, 3]]. Edge indices are zero-based positions in this exact list. A matching contains distinct edge indices with no shared endpoint. Enumerate every edge subset to determine the global maximum size; include the empty matching. List indices within each matching increasingly and enumerate matchings in increasing edge-subset mask order. Maximum,inclusion-maximal,perfect and unique are distinct properties. Give (matching_count,maximum_size,maximum_matchings,exhaustive_all_edge_subsets).
View answer and explanation
(7, 2, [[1, 2], [0, 3]], exhaustive_all_edge_subsets)
Reject every edge subset with a repeated endpoint,then maximize size over all remaining matchings. Count all valid matchings including the empty one and retain every maximum. This finite enumeration is not a general complexity or perfect-matching theorem. The exact result, retaining the requested order, is (7, 2, [[1, 2], [0, 3]], exhaustive_all_edge_subsets).
Reviewed step-by-step derivation
Maximum matching · qf-mod34-maxmatching-maximum-size · candidate
8. The Maximum Size Requires All Valid Matchings
Use the complete bounded simple graph on Fin(6)={0,…,5} with canonical ordered undirected edges=[[0, 1], [0, 2], [0, 3], [1, 2], [2, 3]]. Edge indices are zero-based positions in this exact list. A matching contains distinct edge indices with no shared endpoint. Enumerate every edge subset to determine the global maximum size; include the empty matching. List indices within each matching increasingly and enumerate matchings in increasing edge-subset mask order. Maximum,inclusion-maximal,perfect and unique are distinct properties. Give (matching_count,maximum_size,maximum_matchings,exhaustive_all_edge_subsets).
View answer and explanation
(8, 2, [[2, 3], [0, 4]], exhaustive_all_edge_subsets)
Reject every edge subset with a repeated endpoint,then maximize size over all remaining matchings. Count all valid matchings including the empty one and retain every maximum. This finite enumeration is not a general complexity or perfect-matching theorem. The exact result, retaining the requested order, is (8, 2, [[2, 3], [0, 4]], exhaustive_all_edge_subsets).
Reviewed step-by-step derivation
Maximum matching · qf-mod34-maxmatching-maximum-size · candidate
9. The Maximum Size Requires All Valid Matchings
Use the complete bounded simple graph on Fin(6)={0,…,5} with canonical ordered undirected edges=[[0, 1], [1, 3], [2, 3]]. Edge indices are zero-based positions in this exact list. A matching contains distinct edge indices with no shared endpoint. Enumerate every edge subset to determine the global maximum size; include the empty matching. List indices within each matching increasingly and enumerate matchings in increasing edge-subset mask order. Maximum,inclusion-maximal,perfect and unique are distinct properties. Give (matching_count,maximum_size,maximum_matchings,exhaustive_all_edge_subsets).
View answer and explanation
(5, 2, [[0, 2]], exhaustive_all_edge_subsets)
Reject every edge subset with a repeated endpoint,then maximize size over all remaining matchings. Count all valid matchings including the empty one and retain every maximum. This finite enumeration is not a general complexity or perfect-matching theorem. The exact result, retaining the requested order, is (5, 2, [[0, 2]], exhaustive_all_edge_subsets).
Reviewed step-by-step derivation
Maximum matching · qf-mod34-maxmatching-maximum-size · candidate
10. The Maximum Size Requires All Valid Matchings
Use the complete bounded simple graph on Fin(6)={0,…,5} with canonical ordered undirected edges=[[0, 1], [0, 2], [1, 3], [2, 3]]. Edge indices are zero-based positions in this exact list. A matching contains distinct edge indices with no shared endpoint. Enumerate every edge subset to determine the global maximum size; include the empty matching. List indices within each matching increasingly and enumerate matchings in increasing edge-subset mask order. Maximum,inclusion-maximal,perfect and unique are distinct properties. Give (matching_count,maximum_size,maximum_matchings,exhaustive_all_edge_subsets).
View answer and explanation
(7, 2, [[1, 2], [0, 3]], exhaustive_all_edge_subsets)
Reject every edge subset with a repeated endpoint,then maximize size over all remaining matchings. Count all valid matchings including the empty one and retain every maximum. This finite enumeration is not a general complexity or perfect-matching theorem. The exact result, retaining the requested order, is (7, 2, [[1, 2], [0, 3]], exhaustive_all_edge_subsets).
Reviewed step-by-step derivation
Maximum matching · qf-mod34-maxmatching-maximum-size · candidate
11. The Maximum Size Requires All Valid Matchings
Use the complete bounded simple graph on Fin(6)={0,…,5} with canonical ordered undirected edges=[[0, 1], [0, 3], [1, 3], [2, 3]]. Edge indices are zero-based positions in this exact list. A matching contains distinct edge indices with no shared endpoint. Enumerate every edge subset to determine the global maximum size; include the empty matching. List indices within each matching increasingly and enumerate matchings in increasing edge-subset mask order. Maximum,inclusion-maximal,perfect and unique are distinct properties. Give (matching_count,maximum_size,maximum_matchings,exhaustive_all_edge_subsets).
View answer and explanation
(6, 2, [[0, 3]], exhaustive_all_edge_subsets)
Reject every edge subset with a repeated endpoint,then maximize size over all remaining matchings. Count all valid matchings including the empty one and retain every maximum. This finite enumeration is not a general complexity or perfect-matching theorem. The exact result, retaining the requested order, is (6, 2, [[0, 3]], exhaustive_all_edge_subsets).
Reviewed step-by-step derivation
Maximum matching · qf-mod34-maxmatching-maximum-size · candidate
12. The Maximum Size Requires All Valid Matchings
Use the complete bounded simple graph on Fin(6)={0,…,5} with canonical ordered undirected edges=[[0, 1], [0, 2], [0, 3], [1, 3], [2, 3]]. Edge indices are zero-based positions in this exact list. A matching contains distinct edge indices with no shared endpoint. Enumerate every edge subset to determine the global maximum size; include the empty matching. List indices within each matching increasingly and enumerate matchings in increasing edge-subset mask order. Maximum,inclusion-maximal,perfect and unique are distinct properties. Give (matching_count,maximum_size,maximum_matchings,exhaustive_all_edge_subsets).
View answer and explanation
(8, 2, [[1, 3], [0, 4]], exhaustive_all_edge_subsets)
Reject every edge subset with a repeated endpoint,then maximize size over all remaining matchings. Count all valid matchings including the empty one and retain every maximum. This finite enumeration is not a general complexity or perfect-matching theorem. The exact result, retaining the requested order, is (8, 2, [[1, 3], [0, 4]], exhaustive_all_edge_subsets).
Reviewed step-by-step derivation
Maximum matching · qf-mod34-maxmatching-maximum-size · candidate
13. The Maximum Size Requires All Valid Matchings
Use the complete bounded simple graph on Fin(6)={0,…,5} with canonical ordered undirected edges=[[0, 1], [1, 2], [1, 3], [2, 3]]. Edge indices are zero-based positions in this exact list. A matching contains distinct edge indices with no shared endpoint. Enumerate every edge subset to determine the global maximum size; include the empty matching. List indices within each matching increasingly and enumerate matchings in increasing edge-subset mask order. Maximum,inclusion-maximal,perfect and unique are distinct properties. Give (matching_count,maximum_size,maximum_matchings,exhaustive_all_edge_subsets).
View answer and explanation
(6, 2, [[0, 3]], exhaustive_all_edge_subsets)
Reject every edge subset with a repeated endpoint,then maximize size over all remaining matchings. Count all valid matchings including the empty one and retain every maximum. This finite enumeration is not a general complexity or perfect-matching theorem. The exact result, retaining the requested order, is (6, 2, [[0, 3]], exhaustive_all_edge_subsets).
Reviewed step-by-step derivation
Maximum matching · qf-mod34-maxmatching-maximum-size · candidate
14. The Maximum Size Requires All Valid Matchings
Use the complete bounded simple graph on Fin(6)={0,…,5} with canonical ordered undirected edges=[[0, 1], [0, 2], [1, 2], [1, 3], [2, 3]]. Edge indices are zero-based positions in this exact list. A matching contains distinct edge indices with no shared endpoint. Enumerate every edge subset to determine the global maximum size; include the empty matching. List indices within each matching increasingly and enumerate matchings in increasing edge-subset mask order. Maximum,inclusion-maximal,perfect and unique are distinct properties. Give (matching_count,maximum_size,maximum_matchings,exhaustive_all_edge_subsets).
View answer and explanation
(8, 2, [[1, 3], [0, 4]], exhaustive_all_edge_subsets)
Reject every edge subset with a repeated endpoint,then maximize size over all remaining matchings. Count all valid matchings including the empty one and retain every maximum. This finite enumeration is not a general complexity or perfect-matching theorem. The exact result, retaining the requested order, is (8, 2, [[1, 3], [0, 4]], exhaustive_all_edge_subsets).
Reviewed step-by-step derivation
Maximum matching · qf-mod34-maxmatching-maximum-size · candidate
15. The Maximum Size Requires All Valid Matchings
Use the complete bounded simple graph on Fin(6)={0,…,5} with canonical ordered undirected edges=[[0, 1], [0, 3], [1, 2], [1, 3], [2, 3]]. Edge indices are zero-based positions in this exact list. A matching contains distinct edge indices with no shared endpoint. Enumerate every edge subset to determine the global maximum size; include the empty matching. List indices within each matching increasingly and enumerate matchings in increasing edge-subset mask order. Maximum,inclusion-maximal,perfect and unique are distinct properties. Give (matching_count,maximum_size,maximum_matchings,exhaustive_all_edge_subsets).
View answer and explanation
(8, 2, [[1, 2], [0, 4]], exhaustive_all_edge_subsets)
Reject every edge subset with a repeated endpoint,then maximize size over all remaining matchings. Count all valid matchings including the empty one and retain every maximum. This finite enumeration is not a general complexity or perfect-matching theorem. The exact result, retaining the requested order, is (8, 2, [[1, 2], [0, 4]], exhaustive_all_edge_subsets).
Reviewed step-by-step derivation
Maximum matching · qf-mod34-maxmatching-maximum-size · candidate
16. The Maximum Size Requires All Valid Matchings
Use the complete bounded simple graph on Fin(6)={0,…,5} with canonical ordered undirected edges=[[0, 1], [0, 2], [0, 3], [1, 2], [1, 3], [2, 3]]. Edge indices are zero-based positions in this exact list. A matching contains distinct edge indices with no shared endpoint. Enumerate every edge subset to determine the global maximum size; include the empty matching. List indices within each matching increasingly and enumerate matchings in increasing edge-subset mask order. Maximum,inclusion-maximal,perfect and unique are distinct properties. Give (matching_count,maximum_size,maximum_matchings,exhaustive_all_edge_subsets).
View answer and explanation
(10, 2, [[2, 3], [1, 4], [0, 5]], exhaustive_all_edge_subsets)
Reject every edge subset with a repeated endpoint,then maximize size over all remaining matchings. Count all valid matchings including the empty one and retain every maximum. This finite enumeration is not a general complexity or perfect-matching theorem. The exact result, retaining the requested order, is (10, 2, [[2, 3], [1, 4], [0, 5]], exhaustive_all_edge_subsets).
Reviewed step-by-step derivation
Maximum matching · qf-mod34-maxmatching-maximum-size · candidate
17. The Maximum Size Requires All Valid Matchings
Use the complete bounded simple graph on Fin(6)={0,…,5} with canonical ordered undirected edges=[[0, 1], [2, 3], [4, 5]]. Edge indices are zero-based positions in this exact list. A matching contains distinct edge indices with no shared endpoint. Enumerate every edge subset to determine the global maximum size; include the empty matching. List indices within each matching increasingly and enumerate matchings in increasing edge-subset mask order. Maximum,inclusion-maximal,perfect and unique are distinct properties. Give (matching_count,maximum_size,maximum_matchings,exhaustive_all_edge_subsets).
View answer and explanation
(8, 3, [[0, 1, 2]], exhaustive_all_edge_subsets)
Reject every edge subset with a repeated endpoint,then maximize size over all remaining matchings. Count all valid matchings including the empty one and retain every maximum. This finite enumeration is not a general complexity or perfect-matching theorem. The exact result, retaining the requested order, is (8, 3, [[0, 1, 2]], exhaustive_all_edge_subsets).
Reviewed step-by-step derivation
Maximum matching · qf-mod34-maxmatching-maximum-size · candidate
18. The Maximum Size Requires All Valid Matchings
Use the complete bounded simple graph on Fin(6)={0,…,5} with canonical ordered undirected edges=[[0, 1], [0, 2], [2, 3], [4, 5]]. Edge indices are zero-based positions in this exact list. A matching contains distinct edge indices with no shared endpoint. Enumerate every edge subset to determine the global maximum size; include the empty matching. List indices within each matching increasingly and enumerate matchings in increasing edge-subset mask order. Maximum,inclusion-maximal,perfect and unique are distinct properties. Give (matching_count,maximum_size,maximum_matchings,exhaustive_all_edge_subsets).
View answer and explanation
(10, 3, [[0, 2, 3]], exhaustive_all_edge_subsets)
Reject every edge subset with a repeated endpoint,then maximize size over all remaining matchings. Count all valid matchings including the empty one and retain every maximum. This finite enumeration is not a general complexity or perfect-matching theorem. The exact result, retaining the requested order, is (10, 3, [[0, 2, 3]], exhaustive_all_edge_subsets).
Reviewed step-by-step derivation
Maximum matching · qf-mod34-maxmatching-maximum-size · candidate
19. The Maximum Size Requires All Valid Matchings
Use the complete bounded simple graph on Fin(6)={0,…,5} with canonical ordered undirected edges=[[0, 1], [0, 3], [2, 3], [4, 5]]. Edge indices are zero-based positions in this exact list. A matching contains distinct edge indices with no shared endpoint. Enumerate every edge subset to determine the global maximum size; include the empty matching. List indices within each matching increasingly and enumerate matchings in increasing edge-subset mask order. Maximum,inclusion-maximal,perfect and unique are distinct properties. Give (matching_count,maximum_size,maximum_matchings,exhaustive_all_edge_subsets).
View answer and explanation
(10, 3, [[0, 2, 3]], exhaustive_all_edge_subsets)
Reject every edge subset with a repeated endpoint,then maximize size over all remaining matchings. Count all valid matchings including the empty one and retain every maximum. This finite enumeration is not a general complexity or perfect-matching theorem. The exact result, retaining the requested order, is (10, 3, [[0, 2, 3]], exhaustive_all_edge_subsets).
Reviewed step-by-step derivation
Maximum matching · qf-mod34-maxmatching-maximum-size · candidate
20. The Maximum Size Requires All Valid Matchings
Use the complete bounded simple graph on Fin(6)={0,…,5} with canonical ordered undirected edges=[[0, 1], [0, 2], [0, 3], [2, 3], [4, 5]]. Edge indices are zero-based positions in this exact list. A matching contains distinct edge indices with no shared endpoint. Enumerate every edge subset to determine the global maximum size; include the empty matching. List indices within each matching increasingly and enumerate matchings in increasing edge-subset mask order. Maximum,inclusion-maximal,perfect and unique are distinct properties. Give (matching_count,maximum_size,maximum_matchings,exhaustive_all_edge_subsets).
View answer and explanation
(12, 3, [[0, 3, 4]], exhaustive_all_edge_subsets)
Reject every edge subset with a repeated endpoint,then maximize size over all remaining matchings. Count all valid matchings including the empty one and retain every maximum. This finite enumeration is not a general complexity or perfect-matching theorem. The exact result, retaining the requested order, is (12, 3, [[0, 3, 4]], exhaustive_all_edge_subsets).
Reviewed step-by-step derivation
Maximum matching · qf-mod34-maxmatching-maximum-size · candidate
21. The Maximum Size Requires All Valid Matchings
Use the complete bounded simple graph on Fin(6)={0,…,5} with canonical ordered undirected edges=[[0, 1], [1, 2], [2, 3], [4, 5]]. Edge indices are zero-based positions in this exact list. A matching contains distinct edge indices with no shared endpoint. Enumerate every edge subset to determine the global maximum size; include the empty matching. List indices within each matching increasingly and enumerate matchings in increasing edge-subset mask order. Maximum,inclusion-maximal,perfect and unique are distinct properties. Give (matching_count,maximum_size,maximum_matchings,exhaustive_all_edge_subsets).
View answer and explanation
(10, 3, [[0, 2, 3]], exhaustive_all_edge_subsets)
Reject every edge subset with a repeated endpoint,then maximize size over all remaining matchings. Count all valid matchings including the empty one and retain every maximum. This finite enumeration is not a general complexity or perfect-matching theorem. The exact result, retaining the requested order, is (10, 3, [[0, 2, 3]], exhaustive_all_edge_subsets).
Reviewed step-by-step derivation
Maximum matching · qf-mod34-maxmatching-maximum-size · candidate
22. The Maximum Size Requires All Valid Matchings
Use the complete bounded simple graph on Fin(6)={0,…,5} with canonical ordered undirected edges=[[0, 1], [0, 2], [1, 2], [2, 3], [4, 5]]. Edge indices are zero-based positions in this exact list. A matching contains distinct edge indices with no shared endpoint. Enumerate every edge subset to determine the global maximum size; include the empty matching. List indices within each matching increasingly and enumerate matchings in increasing edge-subset mask order. Maximum,inclusion-maximal,perfect and unique are distinct properties. Give (matching_count,maximum_size,maximum_matchings,exhaustive_all_edge_subsets).
View answer and explanation
(12, 3, [[0, 3, 4]], exhaustive_all_edge_subsets)
Reject every edge subset with a repeated endpoint,then maximize size over all remaining matchings. Count all valid matchings including the empty one and retain every maximum. This finite enumeration is not a general complexity or perfect-matching theorem. The exact result, retaining the requested order, is (12, 3, [[0, 3, 4]], exhaustive_all_edge_subsets).
Reviewed step-by-step derivation
Maximum matching · qf-mod34-maxmatching-maximum-size · candidate
23. The Maximum Size Requires All Valid Matchings
Use the complete bounded simple graph on Fin(6)={0,…,5} with canonical ordered undirected edges=[[0, 1], [0, 3], [1, 2], [2, 3], [4, 5]]. Edge indices are zero-based positions in this exact list. A matching contains distinct edge indices with no shared endpoint. Enumerate every edge subset to determine the global maximum size; include the empty matching. List indices within each matching increasingly and enumerate matchings in increasing edge-subset mask order. Maximum,inclusion-maximal,perfect and unique are distinct properties. Give (matching_count,maximum_size,maximum_matchings,exhaustive_all_edge_subsets).
View answer and explanation
(14, 3, [[1, 2, 4], [0, 3, 4]], exhaustive_all_edge_subsets)
Reject every edge subset with a repeated endpoint,then maximize size over all remaining matchings. Count all valid matchings including the empty one and retain every maximum. This finite enumeration is not a general complexity or perfect-matching theorem. The exact result, retaining the requested order, is (14, 3, [[1, 2, 4], [0, 3, 4]], exhaustive_all_edge_subsets).
Reviewed step-by-step derivation
Maximum matching · qf-mod34-maxmatching-maximum-size · candidate
24. The Maximum Size Requires All Valid Matchings
Use the complete bounded simple graph on Fin(6)={0,…,5} with canonical ordered undirected edges=[[0, 1], [0, 2], [0, 3], [1, 2], [2, 3], [4, 5]]. Edge indices are zero-based positions in this exact list. A matching contains distinct edge indices with no shared endpoint. Enumerate every edge subset to determine the global maximum size; include the empty matching. List indices within each matching increasingly and enumerate matchings in increasing edge-subset mask order. Maximum,inclusion-maximal,perfect and unique are distinct properties. Give (matching_count,maximum_size,maximum_matchings,exhaustive_all_edge_subsets).
View answer and explanation
(16, 3, [[2, 3, 5], [0, 4, 5]], exhaustive_all_edge_subsets)
Reject every edge subset with a repeated endpoint,then maximize size over all remaining matchings. Count all valid matchings including the empty one and retain every maximum. This finite enumeration is not a general complexity or perfect-matching theorem. The exact result, retaining the requested order, is (16, 3, [[2, 3, 5], [0, 4, 5]], exhaustive_all_edge_subsets).
Reviewed step-by-step derivation
Maximum matching · qf-mod34-maxmatching-maximum-size · candidate
25. The Maximum Size Requires All Valid Matchings
Use the complete bounded simple graph on Fin(6)={0,…,5} with canonical ordered undirected edges=[[0, 1], [1, 3], [2, 3], [4, 5]]. Edge indices are zero-based positions in this exact list. A matching contains distinct edge indices with no shared endpoint. Enumerate every edge subset to determine the global maximum size; include the empty matching. List indices within each matching increasingly and enumerate matchings in increasing edge-subset mask order. Maximum,inclusion-maximal,perfect and unique are distinct properties. Give (matching_count,maximum_size,maximum_matchings,exhaustive_all_edge_subsets).
View answer and explanation
(10, 3, [[0, 2, 3]], exhaustive_all_edge_subsets)
Reject every edge subset with a repeated endpoint,then maximize size over all remaining matchings. Count all valid matchings including the empty one and retain every maximum. This finite enumeration is not a general complexity or perfect-matching theorem. The exact result, retaining the requested order, is (10, 3, [[0, 2, 3]], exhaustive_all_edge_subsets).
Reviewed step-by-step derivation
Maximum matching · qf-mod34-maxmatching-maximum-size · candidate
26. Validity Precedes a Maximum-Matching Certificate
Use the complete bounded simple graph on Fin(6)={0,…,5} with canonical ordered undirected edges=[[0, 1], [2, 3]]. Edge indices are zero-based positions in this exact list. A matching contains distinct edge indices with no shared endpoint. Enumerate every edge subset to determine the global maximum size; include the empty matching. List indices within each matching increasingly and enumerate matchings in increasing edge-subset mask order. Maximum,inclusion-maximal,perfect and unique are distinct properties. Proposed edge-index collection=[0, 1]. Give (matching,valid?,size,maximum_size,is_maximum?,maximum_not_necessarily_unique_or_perfect).
View answer and explanation
([0, 1], Yes, 2, 2, Yes, maximum_not_necessarily_unique_or_perfect)
Retain the proposed collection size even if it is invalid. The is_maximum flag is valid AND size equals the global maximum. Size equality alone cannot turn conflicting edges into a matching. The exact result, retaining the requested order, is ([0, 1], Yes, 2, 2, Yes, maximum_not_necessarily_unique_or_perfect).
Reviewed step-by-step derivation
Maximum matching · qf-mod34-maxmatching-maximum-audit · candidate
27. Validity Precedes a Maximum-Matching Certificate
Use the complete bounded simple graph on Fin(6)={0,…,5} with canonical ordered undirected edges=[[0, 1], [0, 2], [2, 3]]. Edge indices are zero-based positions in this exact list. A matching contains distinct edge indices with no shared endpoint. Enumerate every edge subset to determine the global maximum size; include the empty matching. List indices within each matching increasingly and enumerate matchings in increasing edge-subset mask order. Maximum,inclusion-maximal,perfect and unique are distinct properties. Proposed edge-index collection=[0, 1]. Give (matching,valid?,size,maximum_size,is_maximum?,maximum_not_necessarily_unique_or_perfect).
View answer and explanation
([0, 1], No, 2, 2, No, maximum_not_necessarily_unique_or_perfect)
Retain the proposed collection size even if it is invalid. The is_maximum flag is valid AND size equals the global maximum. Size equality alone cannot turn conflicting edges into a matching. The exact result, retaining the requested order, is ([0, 1], No, 2, 2, No, maximum_not_necessarily_unique_or_perfect).
Reviewed step-by-step derivation
Maximum matching · qf-mod34-maxmatching-maximum-audit · candidate
28. Validity Precedes a Maximum-Matching Certificate
Use the complete bounded simple graph on Fin(6)={0,…,5} with canonical ordered undirected edges=[[0, 1], [0, 3], [2, 3]]. Edge indices are zero-based positions in this exact list. A matching contains distinct edge indices with no shared endpoint. Enumerate every edge subset to determine the global maximum size; include the empty matching. List indices within each matching increasingly and enumerate matchings in increasing edge-subset mask order. Maximum,inclusion-maximal,perfect and unique are distinct properties. Proposed edge-index collection=[0, 1]. Give (matching,valid?,size,maximum_size,is_maximum?,maximum_not_necessarily_unique_or_perfect).
View answer and explanation
([0, 1], No, 2, 2, No, maximum_not_necessarily_unique_or_perfect)
Retain the proposed collection size even if it is invalid. The is_maximum flag is valid AND size equals the global maximum. Size equality alone cannot turn conflicting edges into a matching. The exact result, retaining the requested order, is ([0, 1], No, 2, 2, No, maximum_not_necessarily_unique_or_perfect).
Reviewed step-by-step derivation
Maximum matching · qf-mod34-maxmatching-maximum-audit · candidate
29. Validity Precedes a Maximum-Matching Certificate
Use the complete bounded simple graph on Fin(6)={0,…,5} with canonical ordered undirected edges=[[0, 1], [0, 2], [0, 3], [2, 3]]. Edge indices are zero-based positions in this exact list. A matching contains distinct edge indices with no shared endpoint. Enumerate every edge subset to determine the global maximum size; include the empty matching. List indices within each matching increasingly and enumerate matchings in increasing edge-subset mask order. Maximum,inclusion-maximal,perfect and unique are distinct properties. Proposed edge-index collection=[0, 1]. Give (matching,valid?,size,maximum_size,is_maximum?,maximum_not_necessarily_unique_or_perfect).
View answer and explanation
([0, 1], No, 2, 2, No, maximum_not_necessarily_unique_or_perfect)
Retain the proposed collection size even if it is invalid. The is_maximum flag is valid AND size equals the global maximum. Size equality alone cannot turn conflicting edges into a matching. The exact result, retaining the requested order, is ([0, 1], No, 2, 2, No, maximum_not_necessarily_unique_or_perfect).
Reviewed step-by-step derivation
Maximum matching · qf-mod34-maxmatching-maximum-audit · candidate
30. Validity Precedes a Maximum-Matching Certificate
Use the complete bounded simple graph on Fin(6)={0,…,5} with canonical ordered undirected edges=[[0, 1], [1, 2], [2, 3]]. Edge indices are zero-based positions in this exact list. A matching contains distinct edge indices with no shared endpoint. Enumerate every edge subset to determine the global maximum size; include the empty matching. List indices within each matching increasingly and enumerate matchings in increasing edge-subset mask order. Maximum,inclusion-maximal,perfect and unique are distinct properties. Proposed edge-index collection=[0, 1]. Give (matching,valid?,size,maximum_size,is_maximum?,maximum_not_necessarily_unique_or_perfect).
View answer and explanation
([0, 1], No, 2, 2, No, maximum_not_necessarily_unique_or_perfect)
Retain the proposed collection size even if it is invalid. The is_maximum flag is valid AND size equals the global maximum. Size equality alone cannot turn conflicting edges into a matching. The exact result, retaining the requested order, is ([0, 1], No, 2, 2, No, maximum_not_necessarily_unique_or_perfect).
Reviewed step-by-step derivation
Maximum matching · qf-mod34-maxmatching-maximum-audit · candidate
31. Validity Precedes a Maximum-Matching Certificate
Use the complete bounded simple graph on Fin(6)={0,…,5} with canonical ordered undirected edges=[[0, 1], [0, 2], [1, 2], [2, 3]]. Edge indices are zero-based positions in this exact list. A matching contains distinct edge indices with no shared endpoint. Enumerate every edge subset to determine the global maximum size; include the empty matching. List indices within each matching increasingly and enumerate matchings in increasing edge-subset mask order. Maximum,inclusion-maximal,perfect and unique are distinct properties. Proposed edge-index collection=[0, 1]. Give (matching,valid?,size,maximum_size,is_maximum?,maximum_not_necessarily_unique_or_perfect).
View answer and explanation
([0, 1], No, 2, 2, No, maximum_not_necessarily_unique_or_perfect)
Retain the proposed collection size even if it is invalid. The is_maximum flag is valid AND size equals the global maximum. Size equality alone cannot turn conflicting edges into a matching. The exact result, retaining the requested order, is ([0, 1], No, 2, 2, No, maximum_not_necessarily_unique_or_perfect).
Reviewed step-by-step derivation
Maximum matching · qf-mod34-maxmatching-maximum-audit · candidate
32. Validity Precedes a Maximum-Matching Certificate
Use the complete bounded simple graph on Fin(6)={0,…,5} with canonical ordered undirected edges=[[0, 1], [0, 3], [1, 2], [2, 3]]. Edge indices are zero-based positions in this exact list. A matching contains distinct edge indices with no shared endpoint. Enumerate every edge subset to determine the global maximum size; include the empty matching. List indices within each matching increasingly and enumerate matchings in increasing edge-subset mask order. Maximum,inclusion-maximal,perfect and unique are distinct properties. Proposed edge-index collection=[0, 1]. Give (matching,valid?,size,maximum_size,is_maximum?,maximum_not_necessarily_unique_or_perfect).
View answer and explanation
([0, 1], No, 2, 2, No, maximum_not_necessarily_unique_or_perfect)
Retain the proposed collection size even if it is invalid. The is_maximum flag is valid AND size equals the global maximum. Size equality alone cannot turn conflicting edges into a matching. The exact result, retaining the requested order, is ([0, 1], No, 2, 2, No, maximum_not_necessarily_unique_or_perfect).
Reviewed step-by-step derivation
Maximum matching · qf-mod34-maxmatching-maximum-audit · candidate
33. Validity Precedes a Maximum-Matching Certificate
Use the complete bounded simple graph on Fin(6)={0,…,5} with canonical ordered undirected edges=[[0, 1], [0, 2], [0, 3], [1, 2], [2, 3]]. Edge indices are zero-based positions in this exact list. A matching contains distinct edge indices with no shared endpoint. Enumerate every edge subset to determine the global maximum size; include the empty matching. List indices within each matching increasingly and enumerate matchings in increasing edge-subset mask order. Maximum,inclusion-maximal,perfect and unique are distinct properties. Proposed edge-index collection=[0, 1]. Give (matching,valid?,size,maximum_size,is_maximum?,maximum_not_necessarily_unique_or_perfect).
View answer and explanation
([0, 1], No, 2, 2, No, maximum_not_necessarily_unique_or_perfect)
Retain the proposed collection size even if it is invalid. The is_maximum flag is valid AND size equals the global maximum. Size equality alone cannot turn conflicting edges into a matching. The exact result, retaining the requested order, is ([0, 1], No, 2, 2, No, maximum_not_necessarily_unique_or_perfect).
Reviewed step-by-step derivation
Maximum matching · qf-mod34-maxmatching-maximum-audit · candidate
34. Validity Precedes a Maximum-Matching Certificate
Use the complete bounded simple graph on Fin(6)={0,…,5} with canonical ordered undirected edges=[[0, 1], [1, 3], [2, 3]]. Edge indices are zero-based positions in this exact list. A matching contains distinct edge indices with no shared endpoint. Enumerate every edge subset to determine the global maximum size; include the empty matching. List indices within each matching increasingly and enumerate matchings in increasing edge-subset mask order. Maximum,inclusion-maximal,perfect and unique are distinct properties. Proposed edge-index collection=[0, 1]. Give (matching,valid?,size,maximum_size,is_maximum?,maximum_not_necessarily_unique_or_perfect).
View answer and explanation
([0, 1], No, 2, 2, No, maximum_not_necessarily_unique_or_perfect)
Retain the proposed collection size even if it is invalid. The is_maximum flag is valid AND size equals the global maximum. Size equality alone cannot turn conflicting edges into a matching. The exact result, retaining the requested order, is ([0, 1], No, 2, 2, No, maximum_not_necessarily_unique_or_perfect).
Reviewed step-by-step derivation
Maximum matching · qf-mod34-maxmatching-maximum-audit · candidate
35. Validity Precedes a Maximum-Matching Certificate
Use the complete bounded simple graph on Fin(6)={0,…,5} with canonical ordered undirected edges=[[0, 1], [0, 2], [1, 3], [2, 3]]. Edge indices are zero-based positions in this exact list. A matching contains distinct edge indices with no shared endpoint. Enumerate every edge subset to determine the global maximum size; include the empty matching. List indices within each matching increasingly and enumerate matchings in increasing edge-subset mask order. Maximum,inclusion-maximal,perfect and unique are distinct properties. Proposed edge-index collection=[0, 1]. Give (matching,valid?,size,maximum_size,is_maximum?,maximum_not_necessarily_unique_or_perfect).
View answer and explanation
([0, 1], No, 2, 2, No, maximum_not_necessarily_unique_or_perfect)
Retain the proposed collection size even if it is invalid. The is_maximum flag is valid AND size equals the global maximum. Size equality alone cannot turn conflicting edges into a matching. The exact result, retaining the requested order, is ([0, 1], No, 2, 2, No, maximum_not_necessarily_unique_or_perfect).
Reviewed step-by-step derivation
Maximum matching · qf-mod34-maxmatching-maximum-audit · candidate
36. Validity Precedes a Maximum-Matching Certificate
Use the complete bounded simple graph on Fin(6)={0,…,5} with canonical ordered undirected edges=[[0, 1], [0, 3], [1, 3], [2, 3]]. Edge indices are zero-based positions in this exact list. A matching contains distinct edge indices with no shared endpoint. Enumerate every edge subset to determine the global maximum size; include the empty matching. List indices within each matching increasingly and enumerate matchings in increasing edge-subset mask order. Maximum,inclusion-maximal,perfect and unique are distinct properties. Proposed edge-index collection=[0, 1]. Give (matching,valid?,size,maximum_size,is_maximum?,maximum_not_necessarily_unique_or_perfect).
View answer and explanation
([0, 1], No, 2, 2, No, maximum_not_necessarily_unique_or_perfect)
Retain the proposed collection size even if it is invalid. The is_maximum flag is valid AND size equals the global maximum. Size equality alone cannot turn conflicting edges into a matching. The exact result, retaining the requested order, is ([0, 1], No, 2, 2, No, maximum_not_necessarily_unique_or_perfect).
Reviewed step-by-step derivation
Maximum matching · qf-mod34-maxmatching-maximum-audit · candidate
37. Validity Precedes a Maximum-Matching Certificate
Use the complete bounded simple graph on Fin(6)={0,…,5} with canonical ordered undirected edges=[[0, 1], [0, 2], [0, 3], [1, 3], [2, 3]]. Edge indices are zero-based positions in this exact list. A matching contains distinct edge indices with no shared endpoint. Enumerate every edge subset to determine the global maximum size; include the empty matching. List indices within each matching increasingly and enumerate matchings in increasing edge-subset mask order. Maximum,inclusion-maximal,perfect and unique are distinct properties. Proposed edge-index collection=[0, 1]. Give (matching,valid?,size,maximum_size,is_maximum?,maximum_not_necessarily_unique_or_perfect).
View answer and explanation
([0, 1], No, 2, 2, No, maximum_not_necessarily_unique_or_perfect)
Retain the proposed collection size even if it is invalid. The is_maximum flag is valid AND size equals the global maximum. Size equality alone cannot turn conflicting edges into a matching. The exact result, retaining the requested order, is ([0, 1], No, 2, 2, No, maximum_not_necessarily_unique_or_perfect).
Reviewed step-by-step derivation
Maximum matching · qf-mod34-maxmatching-maximum-audit · candidate
38. Validity Precedes a Maximum-Matching Certificate
Use the complete bounded simple graph on Fin(6)={0,…,5} with canonical ordered undirected edges=[[0, 1], [1, 2], [1, 3], [2, 3]]. Edge indices are zero-based positions in this exact list. A matching contains distinct edge indices with no shared endpoint. Enumerate every edge subset to determine the global maximum size; include the empty matching. List indices within each matching increasingly and enumerate matchings in increasing edge-subset mask order. Maximum,inclusion-maximal,perfect and unique are distinct properties. Proposed edge-index collection=[0, 1]. Give (matching,valid?,size,maximum_size,is_maximum?,maximum_not_necessarily_unique_or_perfect).
View answer and explanation
([0, 1], No, 2, 2, No, maximum_not_necessarily_unique_or_perfect)
Retain the proposed collection size even if it is invalid. The is_maximum flag is valid AND size equals the global maximum. Size equality alone cannot turn conflicting edges into a matching. The exact result, retaining the requested order, is ([0, 1], No, 2, 2, No, maximum_not_necessarily_unique_or_perfect).
Reviewed step-by-step derivation
Maximum matching · qf-mod34-maxmatching-maximum-audit · candidate
39. Validity Precedes a Maximum-Matching Certificate
Use the complete bounded simple graph on Fin(6)={0,…,5} with canonical ordered undirected edges=[[0, 1], [0, 2], [1, 2], [1, 3], [2, 3]]. Edge indices are zero-based positions in this exact list. A matching contains distinct edge indices with no shared endpoint. Enumerate every edge subset to determine the global maximum size; include the empty matching. List indices within each matching increasingly and enumerate matchings in increasing edge-subset mask order. Maximum,inclusion-maximal,perfect and unique are distinct properties. Proposed edge-index collection=[0, 1]. Give (matching,valid?,size,maximum_size,is_maximum?,maximum_not_necessarily_unique_or_perfect).
View answer and explanation
([0, 1], No, 2, 2, No, maximum_not_necessarily_unique_or_perfect)
Retain the proposed collection size even if it is invalid. The is_maximum flag is valid AND size equals the global maximum. Size equality alone cannot turn conflicting edges into a matching. The exact result, retaining the requested order, is ([0, 1], No, 2, 2, No, maximum_not_necessarily_unique_or_perfect).
Reviewed step-by-step derivation
Maximum matching · qf-mod34-maxmatching-maximum-audit · candidate
40. Validity Precedes a Maximum-Matching Certificate
Use the complete bounded simple graph on Fin(6)={0,…,5} with canonical ordered undirected edges=[[0, 1], [0, 3], [1, 2], [1, 3], [2, 3]]. Edge indices are zero-based positions in this exact list. A matching contains distinct edge indices with no shared endpoint. Enumerate every edge subset to determine the global maximum size; include the empty matching. List indices within each matching increasingly and enumerate matchings in increasing edge-subset mask order. Maximum,inclusion-maximal,perfect and unique are distinct properties. Proposed edge-index collection=[0, 1]. Give (matching,valid?,size,maximum_size,is_maximum?,maximum_not_necessarily_unique_or_perfect).
View answer and explanation
([0, 1], No, 2, 2, No, maximum_not_necessarily_unique_or_perfect)
Retain the proposed collection size even if it is invalid. The is_maximum flag is valid AND size equals the global maximum. Size equality alone cannot turn conflicting edges into a matching. The exact result, retaining the requested order, is ([0, 1], No, 2, 2, No, maximum_not_necessarily_unique_or_perfect).
Reviewed step-by-step derivation
Maximum matching · qf-mod34-maxmatching-maximum-audit · candidate
41. Validity Precedes a Maximum-Matching Certificate
Use the complete bounded simple graph on Fin(6)={0,…,5} with canonical ordered undirected edges=[[0, 1], [0, 2], [0, 3], [1, 2], [1, 3], [2, 3]]. Edge indices are zero-based positions in this exact list. A matching contains distinct edge indices with no shared endpoint. Enumerate every edge subset to determine the global maximum size; include the empty matching. List indices within each matching increasingly and enumerate matchings in increasing edge-subset mask order. Maximum,inclusion-maximal,perfect and unique are distinct properties. Proposed edge-index collection=[0, 1]. Give (matching,valid?,size,maximum_size,is_maximum?,maximum_not_necessarily_unique_or_perfect).
View answer and explanation
([0, 1], No, 2, 2, No, maximum_not_necessarily_unique_or_perfect)
Retain the proposed collection size even if it is invalid. The is_maximum flag is valid AND size equals the global maximum. Size equality alone cannot turn conflicting edges into a matching. The exact result, retaining the requested order, is ([0, 1], No, 2, 2, No, maximum_not_necessarily_unique_or_perfect).
Reviewed step-by-step derivation
Maximum matching · qf-mod34-maxmatching-maximum-audit · candidate
42. Validity Precedes a Maximum-Matching Certificate
Use the complete bounded simple graph on Fin(6)={0,…,5} with canonical ordered undirected edges=[[0, 1], [2, 3], [4, 5]]. Edge indices are zero-based positions in this exact list. A matching contains distinct edge indices with no shared endpoint. Enumerate every edge subset to determine the global maximum size; include the empty matching. List indices within each matching increasingly and enumerate matchings in increasing edge-subset mask order. Maximum,inclusion-maximal,perfect and unique are distinct properties. Proposed edge-index collection=[0, 1]. Give (matching,valid?,size,maximum_size,is_maximum?,maximum_not_necessarily_unique_or_perfect).
View answer and explanation
([0, 1], Yes, 2, 3, No, maximum_not_necessarily_unique_or_perfect)
Retain the proposed collection size even if it is invalid. The is_maximum flag is valid AND size equals the global maximum. Size equality alone cannot turn conflicting edges into a matching. The exact result, retaining the requested order, is ([0, 1], Yes, 2, 3, No, maximum_not_necessarily_unique_or_perfect).
Reviewed step-by-step derivation
Maximum matching · qf-mod34-maxmatching-maximum-audit · candidate
43. Validity Precedes a Maximum-Matching Certificate
Use the complete bounded simple graph on Fin(6)={0,…,5} with canonical ordered undirected edges=[[0, 1], [0, 2], [2, 3], [4, 5]]. Edge indices are zero-based positions in this exact list. A matching contains distinct edge indices with no shared endpoint. Enumerate every edge subset to determine the global maximum size; include the empty matching. List indices within each matching increasingly and enumerate matchings in increasing edge-subset mask order. Maximum,inclusion-maximal,perfect and unique are distinct properties. Proposed edge-index collection=[0, 1]. Give (matching,valid?,size,maximum_size,is_maximum?,maximum_not_necessarily_unique_or_perfect).
View answer and explanation
([0, 1], No, 2, 3, No, maximum_not_necessarily_unique_or_perfect)
Retain the proposed collection size even if it is invalid. The is_maximum flag is valid AND size equals the global maximum. Size equality alone cannot turn conflicting edges into a matching. The exact result, retaining the requested order, is ([0, 1], No, 2, 3, No, maximum_not_necessarily_unique_or_perfect).
Reviewed step-by-step derivation
Maximum matching · qf-mod34-maxmatching-maximum-audit · candidate
44. Validity Precedes a Maximum-Matching Certificate
Use the complete bounded simple graph on Fin(6)={0,…,5} with canonical ordered undirected edges=[[0, 1], [0, 3], [2, 3], [4, 5]]. Edge indices are zero-based positions in this exact list. A matching contains distinct edge indices with no shared endpoint. Enumerate every edge subset to determine the global maximum size; include the empty matching. List indices within each matching increasingly and enumerate matchings in increasing edge-subset mask order. Maximum,inclusion-maximal,perfect and unique are distinct properties. Proposed edge-index collection=[0, 1]. Give (matching,valid?,size,maximum_size,is_maximum?,maximum_not_necessarily_unique_or_perfect).
View answer and explanation
([0, 1], No, 2, 3, No, maximum_not_necessarily_unique_or_perfect)
Retain the proposed collection size even if it is invalid. The is_maximum flag is valid AND size equals the global maximum. Size equality alone cannot turn conflicting edges into a matching. The exact result, retaining the requested order, is ([0, 1], No, 2, 3, No, maximum_not_necessarily_unique_or_perfect).
Reviewed step-by-step derivation
Maximum matching · qf-mod34-maxmatching-maximum-audit · candidate
45. Validity Precedes a Maximum-Matching Certificate
Use the complete bounded simple graph on Fin(6)={0,…,5} with canonical ordered undirected edges=[[0, 1], [0, 2], [0, 3], [2, 3], [4, 5]]. Edge indices are zero-based positions in this exact list. A matching contains distinct edge indices with no shared endpoint. Enumerate every edge subset to determine the global maximum size; include the empty matching. List indices within each matching increasingly and enumerate matchings in increasing edge-subset mask order. Maximum,inclusion-maximal,perfect and unique are distinct properties. Proposed edge-index collection=[0, 1]. Give (matching,valid?,size,maximum_size,is_maximum?,maximum_not_necessarily_unique_or_perfect).
View answer and explanation
([0, 1], No, 2, 3, No, maximum_not_necessarily_unique_or_perfect)
Retain the proposed collection size even if it is invalid. The is_maximum flag is valid AND size equals the global maximum. Size equality alone cannot turn conflicting edges into a matching. The exact result, retaining the requested order, is ([0, 1], No, 2, 3, No, maximum_not_necessarily_unique_or_perfect).
Reviewed step-by-step derivation
Maximum matching · qf-mod34-maxmatching-maximum-audit · candidate
46. Validity Precedes a Maximum-Matching Certificate
Use the complete bounded simple graph on Fin(6)={0,…,5} with canonical ordered undirected edges=[[0, 1], [1, 2], [2, 3], [4, 5]]. Edge indices are zero-based positions in this exact list. A matching contains distinct edge indices with no shared endpoint. Enumerate every edge subset to determine the global maximum size; include the empty matching. List indices within each matching increasingly and enumerate matchings in increasing edge-subset mask order. Maximum,inclusion-maximal,perfect and unique are distinct properties. Proposed edge-index collection=[0, 1]. Give (matching,valid?,size,maximum_size,is_maximum?,maximum_not_necessarily_unique_or_perfect).
View answer and explanation
([0, 1], No, 2, 3, No, maximum_not_necessarily_unique_or_perfect)
Retain the proposed collection size even if it is invalid. The is_maximum flag is valid AND size equals the global maximum. Size equality alone cannot turn conflicting edges into a matching. The exact result, retaining the requested order, is ([0, 1], No, 2, 3, No, maximum_not_necessarily_unique_or_perfect).
Reviewed step-by-step derivation
Maximum matching · qf-mod34-maxmatching-maximum-audit · candidate
47. Validity Precedes a Maximum-Matching Certificate
Use the complete bounded simple graph on Fin(6)={0,…,5} with canonical ordered undirected edges=[[0, 1], [0, 2], [1, 2], [2, 3], [4, 5]]. Edge indices are zero-based positions in this exact list. A matching contains distinct edge indices with no shared endpoint. Enumerate every edge subset to determine the global maximum size; include the empty matching. List indices within each matching increasingly and enumerate matchings in increasing edge-subset mask order. Maximum,inclusion-maximal,perfect and unique are distinct properties. Proposed edge-index collection=[0, 1]. Give (matching,valid?,size,maximum_size,is_maximum?,maximum_not_necessarily_unique_or_perfect).
View answer and explanation
([0, 1], No, 2, 3, No, maximum_not_necessarily_unique_or_perfect)
Retain the proposed collection size even if it is invalid. The is_maximum flag is valid AND size equals the global maximum. Size equality alone cannot turn conflicting edges into a matching. The exact result, retaining the requested order, is ([0, 1], No, 2, 3, No, maximum_not_necessarily_unique_or_perfect).
Reviewed step-by-step derivation
Maximum matching · qf-mod34-maxmatching-maximum-audit · candidate
48. Validity Precedes a Maximum-Matching Certificate
Use the complete bounded simple graph on Fin(6)={0,…,5} with canonical ordered undirected edges=[[0, 1], [0, 3], [1, 2], [2, 3], [4, 5]]. Edge indices are zero-based positions in this exact list. A matching contains distinct edge indices with no shared endpoint. Enumerate every edge subset to determine the global maximum size; include the empty matching. List indices within each matching increasingly and enumerate matchings in increasing edge-subset mask order. Maximum,inclusion-maximal,perfect and unique are distinct properties. Proposed edge-index collection=[0, 1]. Give (matching,valid?,size,maximum_size,is_maximum?,maximum_not_necessarily_unique_or_perfect).
View answer and explanation
([0, 1], No, 2, 3, No, maximum_not_necessarily_unique_or_perfect)
Retain the proposed collection size even if it is invalid. The is_maximum flag is valid AND size equals the global maximum. Size equality alone cannot turn conflicting edges into a matching. The exact result, retaining the requested order, is ([0, 1], No, 2, 3, No, maximum_not_necessarily_unique_or_perfect).
Reviewed step-by-step derivation
Maximum matching · qf-mod34-maxmatching-maximum-audit · candidate
49. Validity Precedes a Maximum-Matching Certificate
Use the complete bounded simple graph on Fin(6)={0,…,5} with canonical ordered undirected edges=[[0, 1], [0, 2], [0, 3], [1, 2], [2, 3], [4, 5]]. Edge indices are zero-based positions in this exact list. A matching contains distinct edge indices with no shared endpoint. Enumerate every edge subset to determine the global maximum size; include the empty matching. List indices within each matching increasingly and enumerate matchings in increasing edge-subset mask order. Maximum,inclusion-maximal,perfect and unique are distinct properties. Proposed edge-index collection=[0, 1]. Give (matching,valid?,size,maximum_size,is_maximum?,maximum_not_necessarily_unique_or_perfect).
View answer and explanation
([0, 1], No, 2, 3, No, maximum_not_necessarily_unique_or_perfect)
Retain the proposed collection size even if it is invalid. The is_maximum flag is valid AND size equals the global maximum. Size equality alone cannot turn conflicting edges into a matching. The exact result, retaining the requested order, is ([0, 1], No, 2, 3, No, maximum_not_necessarily_unique_or_perfect).
Reviewed step-by-step derivation
Maximum matching · qf-mod34-maxmatching-maximum-audit · candidate
50. Validity Precedes a Maximum-Matching Certificate
Use the complete bounded simple graph on Fin(6)={0,…,5} with canonical ordered undirected edges=[[0, 1], [1, 3], [2, 3], [4, 5]]. Edge indices are zero-based positions in this exact list. A matching contains distinct edge indices with no shared endpoint. Enumerate every edge subset to determine the global maximum size; include the empty matching. List indices within each matching increasingly and enumerate matchings in increasing edge-subset mask order. Maximum,inclusion-maximal,perfect and unique are distinct properties. Proposed edge-index collection=[0, 1]. Give (matching,valid?,size,maximum_size,is_maximum?,maximum_not_necessarily_unique_or_perfect).
View answer and explanation
([0, 1], No, 2, 3, No, maximum_not_necessarily_unique_or_perfect)
Retain the proposed collection size even if it is invalid. The is_maximum flag is valid AND size equals the global maximum. Size equality alone cannot turn conflicting edges into a matching. The exact result, retaining the requested order, is ([0, 1], No, 2, 3, No, maximum_not_necessarily_unique_or_perfect).
Reviewed step-by-step derivation
Maximum matching · qf-mod34-maxmatching-maximum-audit · candidate
51. Distinct Maximum Matchings Need Not Be Perfect
Use the complete bounded simple graph on Fin(6)={0,…,5} with canonical ordered undirected edges=[[0, 1], [2, 3]]. Edge indices are zero-based positions in this exact list. A matching contains distinct edge indices with no shared endpoint. Enumerate every edge subset to determine the global maximum size; include the empty matching. List indices within each matching increasingly and enumerate matchings in increasing edge-subset mask order. Maximum,inclusion-maximal,perfect and unique are distinct properties. Give (maximum_size,maximum_matchings,count,multiple?), listing every maximum in the declared order.
View answer and explanation
(2, [[0, 1]], 1, No)
Exhaustively collect valid matchings of global maximum size and count them. multiple means more than one such matching,not more than one edge or a perfect-matching certificate. The exact result, retaining the requested order, is (2, [[0, 1]], 1, No).
Reviewed step-by-step derivation
Maximum matching · qf-mod34-maxmatching-multiple-maxima · candidate
52. Distinct Maximum Matchings Need Not Be Perfect
Use the complete bounded simple graph on Fin(6)={0,…,5} with canonical ordered undirected edges=[[0, 1], [0, 2], [2, 3]]. Edge indices are zero-based positions in this exact list. A matching contains distinct edge indices with no shared endpoint. Enumerate every edge subset to determine the global maximum size; include the empty matching. List indices within each matching increasingly and enumerate matchings in increasing edge-subset mask order. Maximum,inclusion-maximal,perfect and unique are distinct properties. Give (maximum_size,maximum_matchings,count,multiple?), listing every maximum in the declared order.
View answer and explanation
(2, [[0, 2]], 1, No)
Exhaustively collect valid matchings of global maximum size and count them. multiple means more than one such matching,not more than one edge or a perfect-matching certificate. The exact result, retaining the requested order, is (2, [[0, 2]], 1, No).
Reviewed step-by-step derivation
Maximum matching · qf-mod34-maxmatching-multiple-maxima · candidate
53. Distinct Maximum Matchings Need Not Be Perfect
Use the complete bounded simple graph on Fin(6)={0,…,5} with canonical ordered undirected edges=[[0, 1], [0, 3], [2, 3]]. Edge indices are zero-based positions in this exact list. A matching contains distinct edge indices with no shared endpoint. Enumerate every edge subset to determine the global maximum size; include the empty matching. List indices within each matching increasingly and enumerate matchings in increasing edge-subset mask order. Maximum,inclusion-maximal,perfect and unique are distinct properties. Give (maximum_size,maximum_matchings,count,multiple?), listing every maximum in the declared order.
View answer and explanation
(2, [[0, 2]], 1, No)
Exhaustively collect valid matchings of global maximum size and count them. multiple means more than one such matching,not more than one edge or a perfect-matching certificate. The exact result, retaining the requested order, is (2, [[0, 2]], 1, No).
Reviewed step-by-step derivation
Maximum matching · qf-mod34-maxmatching-multiple-maxima · candidate
54. Distinct Maximum Matchings Need Not Be Perfect
Use the complete bounded simple graph on Fin(6)={0,…,5} with canonical ordered undirected edges=[[0, 1], [0, 2], [0, 3], [2, 3]]. Edge indices are zero-based positions in this exact list. A matching contains distinct edge indices with no shared endpoint. Enumerate every edge subset to determine the global maximum size; include the empty matching. List indices within each matching increasingly and enumerate matchings in increasing edge-subset mask order. Maximum,inclusion-maximal,perfect and unique are distinct properties. Give (maximum_size,maximum_matchings,count,multiple?), listing every maximum in the declared order.
View answer and explanation
(2, [[0, 3]], 1, No)
Exhaustively collect valid matchings of global maximum size and count them. multiple means more than one such matching,not more than one edge or a perfect-matching certificate. The exact result, retaining the requested order, is (2, [[0, 3]], 1, No).
Reviewed step-by-step derivation
Maximum matching · qf-mod34-maxmatching-multiple-maxima · candidate
55. Distinct Maximum Matchings Need Not Be Perfect
Use the complete bounded simple graph on Fin(6)={0,…,5} with canonical ordered undirected edges=[[0, 1], [1, 2], [2, 3]]. Edge indices are zero-based positions in this exact list. A matching contains distinct edge indices with no shared endpoint. Enumerate every edge subset to determine the global maximum size; include the empty matching. List indices within each matching increasingly and enumerate matchings in increasing edge-subset mask order. Maximum,inclusion-maximal,perfect and unique are distinct properties. Give (maximum_size,maximum_matchings,count,multiple?), listing every maximum in the declared order.
View answer and explanation
(2, [[0, 2]], 1, No)
Exhaustively collect valid matchings of global maximum size and count them. multiple means more than one such matching,not more than one edge or a perfect-matching certificate. The exact result, retaining the requested order, is (2, [[0, 2]], 1, No).
Reviewed step-by-step derivation
Maximum matching · qf-mod34-maxmatching-multiple-maxima · candidate
56. Distinct Maximum Matchings Need Not Be Perfect
Use the complete bounded simple graph on Fin(6)={0,…,5} with canonical ordered undirected edges=[[0, 1], [0, 2], [1, 2], [2, 3]]. Edge indices are zero-based positions in this exact list. A matching contains distinct edge indices with no shared endpoint. Enumerate every edge subset to determine the global maximum size; include the empty matching. List indices within each matching increasingly and enumerate matchings in increasing edge-subset mask order. Maximum,inclusion-maximal,perfect and unique are distinct properties. Give (maximum_size,maximum_matchings,count,multiple?), listing every maximum in the declared order.
View answer and explanation
(2, [[0, 3]], 1, No)
Exhaustively collect valid matchings of global maximum size and count them. multiple means more than one such matching,not more than one edge or a perfect-matching certificate. The exact result, retaining the requested order, is (2, [[0, 3]], 1, No).
Reviewed step-by-step derivation
Maximum matching · qf-mod34-maxmatching-multiple-maxima · candidate
57. Distinct Maximum Matchings Need Not Be Perfect
Use the complete bounded simple graph on Fin(6)={0,…,5} with canonical ordered undirected edges=[[0, 1], [0, 3], [1, 2], [2, 3]]. Edge indices are zero-based positions in this exact list. A matching contains distinct edge indices with no shared endpoint. Enumerate every edge subset to determine the global maximum size; include the empty matching. List indices within each matching increasingly and enumerate matchings in increasing edge-subset mask order. Maximum,inclusion-maximal,perfect and unique are distinct properties. Give (maximum_size,maximum_matchings,count,multiple?), listing every maximum in the declared order.
View answer and explanation
(2, [[1, 2], [0, 3]], 2, Yes)
Exhaustively collect valid matchings of global maximum size and count them. multiple means more than one such matching,not more than one edge or a perfect-matching certificate. The exact result, retaining the requested order, is (2, [[1, 2], [0, 3]], 2, Yes).
Reviewed step-by-step derivation
Maximum matching · qf-mod34-maxmatching-multiple-maxima · candidate
58. Distinct Maximum Matchings Need Not Be Perfect
Use the complete bounded simple graph on Fin(6)={0,…,5} with canonical ordered undirected edges=[[0, 1], [0, 2], [0, 3], [1, 2], [2, 3]]. Edge indices are zero-based positions in this exact list. A matching contains distinct edge indices with no shared endpoint. Enumerate every edge subset to determine the global maximum size; include the empty matching. List indices within each matching increasingly and enumerate matchings in increasing edge-subset mask order. Maximum,inclusion-maximal,perfect and unique are distinct properties. Give (maximum_size,maximum_matchings,count,multiple?), listing every maximum in the declared order.
View answer and explanation
(2, [[2, 3], [0, 4]], 2, Yes)
Exhaustively collect valid matchings of global maximum size and count them. multiple means more than one such matching,not more than one edge or a perfect-matching certificate. The exact result, retaining the requested order, is (2, [[2, 3], [0, 4]], 2, Yes).
Reviewed step-by-step derivation
Maximum matching · qf-mod34-maxmatching-multiple-maxima · candidate
59. Distinct Maximum Matchings Need Not Be Perfect
Use the complete bounded simple graph on Fin(6)={0,…,5} with canonical ordered undirected edges=[[0, 1], [1, 3], [2, 3]]. Edge indices are zero-based positions in this exact list. A matching contains distinct edge indices with no shared endpoint. Enumerate every edge subset to determine the global maximum size; include the empty matching. List indices within each matching increasingly and enumerate matchings in increasing edge-subset mask order. Maximum,inclusion-maximal,perfect and unique are distinct properties. Give (maximum_size,maximum_matchings,count,multiple?), listing every maximum in the declared order.
View answer and explanation
(2, [[0, 2]], 1, No)
Exhaustively collect valid matchings of global maximum size and count them. multiple means more than one such matching,not more than one edge or a perfect-matching certificate. The exact result, retaining the requested order, is (2, [[0, 2]], 1, No).
Reviewed step-by-step derivation
Maximum matching · qf-mod34-maxmatching-multiple-maxima · candidate
60. Distinct Maximum Matchings Need Not Be Perfect
Use the complete bounded simple graph on Fin(6)={0,…,5} with canonical ordered undirected edges=[[0, 1], [0, 2], [1, 3], [2, 3]]. Edge indices are zero-based positions in this exact list. A matching contains distinct edge indices with no shared endpoint. Enumerate every edge subset to determine the global maximum size; include the empty matching. List indices within each matching increasingly and enumerate matchings in increasing edge-subset mask order. Maximum,inclusion-maximal,perfect and unique are distinct properties. Give (maximum_size,maximum_matchings,count,multiple?), listing every maximum in the declared order.
View answer and explanation
(2, [[1, 2], [0, 3]], 2, Yes)
Exhaustively collect valid matchings of global maximum size and count them. multiple means more than one such matching,not more than one edge or a perfect-matching certificate. The exact result, retaining the requested order, is (2, [[1, 2], [0, 3]], 2, Yes).
Reviewed step-by-step derivation
Maximum matching · qf-mod34-maxmatching-multiple-maxima · candidate
61. Distinct Maximum Matchings Need Not Be Perfect
Use the complete bounded simple graph on Fin(6)={0,…,5} with canonical ordered undirected edges=[[0, 1], [0, 3], [1, 3], [2, 3]]. Edge indices are zero-based positions in this exact list. A matching contains distinct edge indices with no shared endpoint. Enumerate every edge subset to determine the global maximum size; include the empty matching. List indices within each matching increasingly and enumerate matchings in increasing edge-subset mask order. Maximum,inclusion-maximal,perfect and unique are distinct properties. Give (maximum_size,maximum_matchings,count,multiple?), listing every maximum in the declared order.
View answer and explanation
(2, [[0, 3]], 1, No)
Exhaustively collect valid matchings of global maximum size and count them. multiple means more than one such matching,not more than one edge or a perfect-matching certificate. The exact result, retaining the requested order, is (2, [[0, 3]], 1, No).
Reviewed step-by-step derivation
Maximum matching · qf-mod34-maxmatching-multiple-maxima · candidate
62. Distinct Maximum Matchings Need Not Be Perfect
Use the complete bounded simple graph on Fin(6)={0,…,5} with canonical ordered undirected edges=[[0, 1], [0, 2], [0, 3], [1, 3], [2, 3]]. Edge indices are zero-based positions in this exact list. A matching contains distinct edge indices with no shared endpoint. Enumerate every edge subset to determine the global maximum size; include the empty matching. List indices within each matching increasingly and enumerate matchings in increasing edge-subset mask order. Maximum,inclusion-maximal,perfect and unique are distinct properties. Give (maximum_size,maximum_matchings,count,multiple?), listing every maximum in the declared order.
View answer and explanation
(2, [[1, 3], [0, 4]], 2, Yes)
Exhaustively collect valid matchings of global maximum size and count them. multiple means more than one such matching,not more than one edge or a perfect-matching certificate. The exact result, retaining the requested order, is (2, [[1, 3], [0, 4]], 2, Yes).
Reviewed step-by-step derivation
Maximum matching · qf-mod34-maxmatching-multiple-maxima · candidate
63. Distinct Maximum Matchings Need Not Be Perfect
Use the complete bounded simple graph on Fin(6)={0,…,5} with canonical ordered undirected edges=[[0, 1], [1, 2], [1, 3], [2, 3]]. Edge indices are zero-based positions in this exact list. A matching contains distinct edge indices with no shared endpoint. Enumerate every edge subset to determine the global maximum size; include the empty matching. List indices within each matching increasingly and enumerate matchings in increasing edge-subset mask order. Maximum,inclusion-maximal,perfect and unique are distinct properties. Give (maximum_size,maximum_matchings,count,multiple?), listing every maximum in the declared order.
View answer and explanation
(2, [[0, 3]], 1, No)
Exhaustively collect valid matchings of global maximum size and count them. multiple means more than one such matching,not more than one edge or a perfect-matching certificate. The exact result, retaining the requested order, is (2, [[0, 3]], 1, No).
Reviewed step-by-step derivation
Maximum matching · qf-mod34-maxmatching-multiple-maxima · candidate
64. Distinct Maximum Matchings Need Not Be Perfect
Use the complete bounded simple graph on Fin(6)={0,…,5} with canonical ordered undirected edges=[[0, 1], [0, 2], [1, 2], [1, 3], [2, 3]]. Edge indices are zero-based positions in this exact list. A matching contains distinct edge indices with no shared endpoint. Enumerate every edge subset to determine the global maximum size; include the empty matching. List indices within each matching increasingly and enumerate matchings in increasing edge-subset mask order. Maximum,inclusion-maximal,perfect and unique are distinct properties. Give (maximum_size,maximum_matchings,count,multiple?), listing every maximum in the declared order.
View answer and explanation
(2, [[1, 3], [0, 4]], 2, Yes)
Exhaustively collect valid matchings of global maximum size and count them. multiple means more than one such matching,not more than one edge or a perfect-matching certificate. The exact result, retaining the requested order, is (2, [[1, 3], [0, 4]], 2, Yes).
Reviewed step-by-step derivation
Maximum matching · qf-mod34-maxmatching-multiple-maxima · candidate
65. Distinct Maximum Matchings Need Not Be Perfect
Use the complete bounded simple graph on Fin(6)={0,…,5} with canonical ordered undirected edges=[[0, 1], [0, 3], [1, 2], [1, 3], [2, 3]]. Edge indices are zero-based positions in this exact list. A matching contains distinct edge indices with no shared endpoint. Enumerate every edge subset to determine the global maximum size; include the empty matching. List indices within each matching increasingly and enumerate matchings in increasing edge-subset mask order. Maximum,inclusion-maximal,perfect and unique are distinct properties. Give (maximum_size,maximum_matchings,count,multiple?), listing every maximum in the declared order.
View answer and explanation
(2, [[1, 2], [0, 4]], 2, Yes)
Exhaustively collect valid matchings of global maximum size and count them. multiple means more than one such matching,not more than one edge or a perfect-matching certificate. The exact result, retaining the requested order, is (2, [[1, 2], [0, 4]], 2, Yes).
Reviewed step-by-step derivation
Maximum matching · qf-mod34-maxmatching-multiple-maxima · candidate
66. Distinct Maximum Matchings Need Not Be Perfect
Use the complete bounded simple graph on Fin(6)={0,…,5} with canonical ordered undirected edges=[[0, 1], [0, 2], [0, 3], [1, 2], [1, 3], [2, 3]]. Edge indices are zero-based positions in this exact list. A matching contains distinct edge indices with no shared endpoint. Enumerate every edge subset to determine the global maximum size; include the empty matching. List indices within each matching increasingly and enumerate matchings in increasing edge-subset mask order. Maximum,inclusion-maximal,perfect and unique are distinct properties. Give (maximum_size,maximum_matchings,count,multiple?), listing every maximum in the declared order.
View answer and explanation
(2, [[2, 3], [1, 4], [0, 5]], 3, Yes)
Exhaustively collect valid matchings of global maximum size and count them. multiple means more than one such matching,not more than one edge or a perfect-matching certificate. The exact result, retaining the requested order, is (2, [[2, 3], [1, 4], [0, 5]], 3, Yes).
Reviewed step-by-step derivation
Maximum matching · qf-mod34-maxmatching-multiple-maxima · candidate
67. Distinct Maximum Matchings Need Not Be Perfect
Use the complete bounded simple graph on Fin(6)={0,…,5} with canonical ordered undirected edges=[[0, 1], [2, 3], [4, 5]]. Edge indices are zero-based positions in this exact list. A matching contains distinct edge indices with no shared endpoint. Enumerate every edge subset to determine the global maximum size; include the empty matching. List indices within each matching increasingly and enumerate matchings in increasing edge-subset mask order. Maximum,inclusion-maximal,perfect and unique are distinct properties. Give (maximum_size,maximum_matchings,count,multiple?), listing every maximum in the declared order.
View answer and explanation
(3, [[0, 1, 2]], 1, No)
Exhaustively collect valid matchings of global maximum size and count them. multiple means more than one such matching,not more than one edge or a perfect-matching certificate. The exact result, retaining the requested order, is (3, [[0, 1, 2]], 1, No).
Reviewed step-by-step derivation
Maximum matching · qf-mod34-maxmatching-multiple-maxima · candidate
68. Distinct Maximum Matchings Need Not Be Perfect
Use the complete bounded simple graph on Fin(6)={0,…,5} with canonical ordered undirected edges=[[0, 1], [0, 2], [2, 3], [4, 5]]. Edge indices are zero-based positions in this exact list. A matching contains distinct edge indices with no shared endpoint. Enumerate every edge subset to determine the global maximum size; include the empty matching. List indices within each matching increasingly and enumerate matchings in increasing edge-subset mask order. Maximum,inclusion-maximal,perfect and unique are distinct properties. Give (maximum_size,maximum_matchings,count,multiple?), listing every maximum in the declared order.
View answer and explanation
(3, [[0, 2, 3]], 1, No)
Exhaustively collect valid matchings of global maximum size and count them. multiple means more than one such matching,not more than one edge or a perfect-matching certificate. The exact result, retaining the requested order, is (3, [[0, 2, 3]], 1, No).
Reviewed step-by-step derivation
Maximum matching · qf-mod34-maxmatching-multiple-maxima · candidate
69. Distinct Maximum Matchings Need Not Be Perfect
Use the complete bounded simple graph on Fin(6)={0,…,5} with canonical ordered undirected edges=[[0, 1], [0, 3], [2, 3], [4, 5]]. Edge indices are zero-based positions in this exact list. A matching contains distinct edge indices with no shared endpoint. Enumerate every edge subset to determine the global maximum size; include the empty matching. List indices within each matching increasingly and enumerate matchings in increasing edge-subset mask order. Maximum,inclusion-maximal,perfect and unique are distinct properties. Give (maximum_size,maximum_matchings,count,multiple?), listing every maximum in the declared order.
View answer and explanation
(3, [[0, 2, 3]], 1, No)
Exhaustively collect valid matchings of global maximum size and count them. multiple means more than one such matching,not more than one edge or a perfect-matching certificate. The exact result, retaining the requested order, is (3, [[0, 2, 3]], 1, No).
Reviewed step-by-step derivation
Maximum matching · qf-mod34-maxmatching-multiple-maxima · candidate
70. Distinct Maximum Matchings Need Not Be Perfect
Use the complete bounded simple graph on Fin(6)={0,…,5} with canonical ordered undirected edges=[[0, 1], [0, 2], [0, 3], [2, 3], [4, 5]]. Edge indices are zero-based positions in this exact list. A matching contains distinct edge indices with no shared endpoint. Enumerate every edge subset to determine the global maximum size; include the empty matching. List indices within each matching increasingly and enumerate matchings in increasing edge-subset mask order. Maximum,inclusion-maximal,perfect and unique are distinct properties. Give (maximum_size,maximum_matchings,count,multiple?), listing every maximum in the declared order.
View answer and explanation
(3, [[0, 3, 4]], 1, No)
Exhaustively collect valid matchings of global maximum size and count them. multiple means more than one such matching,not more than one edge or a perfect-matching certificate. The exact result, retaining the requested order, is (3, [[0, 3, 4]], 1, No).
Reviewed step-by-step derivation
Maximum matching · qf-mod34-maxmatching-multiple-maxima · candidate
71. Distinct Maximum Matchings Need Not Be Perfect
Use the complete bounded simple graph on Fin(6)={0,…,5} with canonical ordered undirected edges=[[0, 1], [1, 2], [2, 3], [4, 5]]. Edge indices are zero-based positions in this exact list. A matching contains distinct edge indices with no shared endpoint. Enumerate every edge subset to determine the global maximum size; include the empty matching. List indices within each matching increasingly and enumerate matchings in increasing edge-subset mask order. Maximum,inclusion-maximal,perfect and unique are distinct properties. Give (maximum_size,maximum_matchings,count,multiple?), listing every maximum in the declared order.
View answer and explanation
(3, [[0, 2, 3]], 1, No)
Exhaustively collect valid matchings of global maximum size and count them. multiple means more than one such matching,not more than one edge or a perfect-matching certificate. The exact result, retaining the requested order, is (3, [[0, 2, 3]], 1, No).
Reviewed step-by-step derivation
Maximum matching · qf-mod34-maxmatching-multiple-maxima · candidate
72. Distinct Maximum Matchings Need Not Be Perfect
Use the complete bounded simple graph on Fin(6)={0,…,5} with canonical ordered undirected edges=[[0, 1], [0, 2], [1, 2], [2, 3], [4, 5]]. Edge indices are zero-based positions in this exact list. A matching contains distinct edge indices with no shared endpoint. Enumerate every edge subset to determine the global maximum size; include the empty matching. List indices within each matching increasingly and enumerate matchings in increasing edge-subset mask order. Maximum,inclusion-maximal,perfect and unique are distinct properties. Give (maximum_size,maximum_matchings,count,multiple?), listing every maximum in the declared order.
View answer and explanation
(3, [[0, 3, 4]], 1, No)
Exhaustively collect valid matchings of global maximum size and count them. multiple means more than one such matching,not more than one edge or a perfect-matching certificate. The exact result, retaining the requested order, is (3, [[0, 3, 4]], 1, No).
Reviewed step-by-step derivation
Maximum matching · qf-mod34-maxmatching-multiple-maxima · candidate
73. Distinct Maximum Matchings Need Not Be Perfect
Use the complete bounded simple graph on Fin(6)={0,…,5} with canonical ordered undirected edges=[[0, 1], [0, 3], [1, 2], [2, 3], [4, 5]]. Edge indices are zero-based positions in this exact list. A matching contains distinct edge indices with no shared endpoint. Enumerate every edge subset to determine the global maximum size; include the empty matching. List indices within each matching increasingly and enumerate matchings in increasing edge-subset mask order. Maximum,inclusion-maximal,perfect and unique are distinct properties. Give (maximum_size,maximum_matchings,count,multiple?), listing every maximum in the declared order.
View answer and explanation
(3, [[1, 2, 4], [0, 3, 4]], 2, Yes)
Exhaustively collect valid matchings of global maximum size and count them. multiple means more than one such matching,not more than one edge or a perfect-matching certificate. The exact result, retaining the requested order, is (3, [[1, 2, 4], [0, 3, 4]], 2, Yes).
Reviewed step-by-step derivation
Maximum matching · qf-mod34-maxmatching-multiple-maxima · candidate
74. Distinct Maximum Matchings Need Not Be Perfect
Use the complete bounded simple graph on Fin(6)={0,…,5} with canonical ordered undirected edges=[[0, 1], [0, 2], [0, 3], [1, 2], [2, 3], [4, 5]]. Edge indices are zero-based positions in this exact list. A matching contains distinct edge indices with no shared endpoint. Enumerate every edge subset to determine the global maximum size; include the empty matching. List indices within each matching increasingly and enumerate matchings in increasing edge-subset mask order. Maximum,inclusion-maximal,perfect and unique are distinct properties. Give (maximum_size,maximum_matchings,count,multiple?), listing every maximum in the declared order.
View answer and explanation
(3, [[2, 3, 5], [0, 4, 5]], 2, Yes)
Exhaustively collect valid matchings of global maximum size and count them. multiple means more than one such matching,not more than one edge or a perfect-matching certificate. The exact result, retaining the requested order, is (3, [[2, 3, 5], [0, 4, 5]], 2, Yes).
Reviewed step-by-step derivation
Maximum matching · qf-mod34-maxmatching-multiple-maxima · candidate
75. Distinct Maximum Matchings Need Not Be Perfect
Use the complete bounded simple graph on Fin(6)={0,…,5} with canonical ordered undirected edges=[[0, 1], [1, 3], [2, 3], [4, 5]]. Edge indices are zero-based positions in this exact list. A matching contains distinct edge indices with no shared endpoint. Enumerate every edge subset to determine the global maximum size; include the empty matching. List indices within each matching increasingly and enumerate matchings in increasing edge-subset mask order. Maximum,inclusion-maximal,perfect and unique are distinct properties. Give (maximum_size,maximum_matchings,count,multiple?), listing every maximum in the declared order.
View answer and explanation
(3, [[0, 2, 3]], 1, No)
Exhaustively collect valid matchings of global maximum size and count them. multiple means more than one such matching,not more than one edge or a perfect-matching certificate. The exact result, retaining the requested order, is (3, [[0, 2, 3]], 1, No).
Reviewed step-by-step derivation
Maximum matching · qf-mod34-maxmatching-multiple-maxima · candidate
76. Conjoin Validity Separately with Maximality and Maximum Size
Use the complete bounded simple graph on Fin(6)={0,…,5} with canonical ordered undirected edges=[[0, 1], [2, 3]]. Edge indices are zero-based positions in this exact list. A matching contains distinct edge indices with no shared endpoint. Enumerate every edge subset to determine the global maximum size; include the empty matching. List indices within each matching increasingly and enumerate matchings in increasing edge-subset mask order. Maximum,inclusion-maximal,perfect and unique are distinct properties. Proposed edge-index collection=[0, 1]. Give (matching,valid?,addable_edges,maximal?,maximum_size,maximum?,maximal_and_maximum_are_distinct_properties). List addable unselected edges increasingly when their endpoints avoid every endpoint used by the proposed collection.
View answer and explanation
([0, 1], Yes, [], Yes, 2, Yes, maximal_and_maximum_are_distinct_properties)
maximal is valid AND no single unused edge can be added. maximum is valid AND the collection size equals the global optimum over all valid matchings. An invalid collection receives neither certificate,even if no edge is addable or its numeric size equals the optimum. The exact result, retaining the requested order, is ([0, 1], Yes, [], Yes, 2, Yes, maximal_and_maximum_are_distinct_properties).
Reviewed step-by-step derivation
Maximum matching · qf-mod34-maxmatching-maximal-vs-maximum · candidate
77. Conjoin Validity Separately with Maximality and Maximum Size
Use the complete bounded simple graph on Fin(6)={0,…,5} with canonical ordered undirected edges=[[0, 1], [0, 2], [2, 3]]. Edge indices are zero-based positions in this exact list. A matching contains distinct edge indices with no shared endpoint. Enumerate every edge subset to determine the global maximum size; include the empty matching. List indices within each matching increasingly and enumerate matchings in increasing edge-subset mask order. Maximum,inclusion-maximal,perfect and unique are distinct properties. Proposed edge-index collection=[0, 1]. Give (matching,valid?,addable_edges,maximal?,maximum_size,maximum?,maximal_and_maximum_are_distinct_properties). List addable unselected edges increasingly when their endpoints avoid every endpoint used by the proposed collection.
View answer and explanation
([0, 1], No, [], No, 2, No, maximal_and_maximum_are_distinct_properties)
maximal is valid AND no single unused edge can be added. maximum is valid AND the collection size equals the global optimum over all valid matchings. An invalid collection receives neither certificate,even if no edge is addable or its numeric size equals the optimum. The exact result, retaining the requested order, is ([0, 1], No, [], No, 2, No, maximal_and_maximum_are_distinct_properties).
Reviewed step-by-step derivation
Maximum matching · qf-mod34-maxmatching-maximal-vs-maximum · candidate
78. Conjoin Validity Separately with Maximality and Maximum Size
Use the complete bounded simple graph on Fin(6)={0,…,5} with canonical ordered undirected edges=[[0, 1], [0, 3], [2, 3]]. Edge indices are zero-based positions in this exact list. A matching contains distinct edge indices with no shared endpoint. Enumerate every edge subset to determine the global maximum size; include the empty matching. List indices within each matching increasingly and enumerate matchings in increasing edge-subset mask order. Maximum,inclusion-maximal,perfect and unique are distinct properties. Proposed edge-index collection=[0, 1]. Give (matching,valid?,addable_edges,maximal?,maximum_size,maximum?,maximal_and_maximum_are_distinct_properties). List addable unselected edges increasingly when their endpoints avoid every endpoint used by the proposed collection.
View answer and explanation
([0, 1], No, [], No, 2, No, maximal_and_maximum_are_distinct_properties)
maximal is valid AND no single unused edge can be added. maximum is valid AND the collection size equals the global optimum over all valid matchings. An invalid collection receives neither certificate,even if no edge is addable or its numeric size equals the optimum. The exact result, retaining the requested order, is ([0, 1], No, [], No, 2, No, maximal_and_maximum_are_distinct_properties).
Reviewed step-by-step derivation
Maximum matching · qf-mod34-maxmatching-maximal-vs-maximum · candidate
79. Conjoin Validity Separately with Maximality and Maximum Size
Use the complete bounded simple graph on Fin(6)={0,…,5} with canonical ordered undirected edges=[[0, 1], [0, 2], [0, 3], [2, 3]]. Edge indices are zero-based positions in this exact list. A matching contains distinct edge indices with no shared endpoint. Enumerate every edge subset to determine the global maximum size; include the empty matching. List indices within each matching increasingly and enumerate matchings in increasing edge-subset mask order. Maximum,inclusion-maximal,perfect and unique are distinct properties. Proposed edge-index collection=[0, 1]. Give (matching,valid?,addable_edges,maximal?,maximum_size,maximum?,maximal_and_maximum_are_distinct_properties). List addable unselected edges increasingly when their endpoints avoid every endpoint used by the proposed collection.
View answer and explanation
([0, 1], No, [], No, 2, No, maximal_and_maximum_are_distinct_properties)
maximal is valid AND no single unused edge can be added. maximum is valid AND the collection size equals the global optimum over all valid matchings. An invalid collection receives neither certificate,even if no edge is addable or its numeric size equals the optimum. The exact result, retaining the requested order, is ([0, 1], No, [], No, 2, No, maximal_and_maximum_are_distinct_properties).
Reviewed step-by-step derivation
Maximum matching · qf-mod34-maxmatching-maximal-vs-maximum · candidate
80. Conjoin Validity Separately with Maximality and Maximum Size
Use the complete bounded simple graph on Fin(6)={0,…,5} with canonical ordered undirected edges=[[0, 1], [1, 2], [2, 3]]. Edge indices are zero-based positions in this exact list. A matching contains distinct edge indices with no shared endpoint. Enumerate every edge subset to determine the global maximum size; include the empty matching. List indices within each matching increasingly and enumerate matchings in increasing edge-subset mask order. Maximum,inclusion-maximal,perfect and unique are distinct properties. Proposed edge-index collection=[0, 1]. Give (matching,valid?,addable_edges,maximal?,maximum_size,maximum?,maximal_and_maximum_are_distinct_properties). List addable unselected edges increasingly when their endpoints avoid every endpoint used by the proposed collection.
View answer and explanation
([0, 1], No, [], No, 2, No, maximal_and_maximum_are_distinct_properties)
maximal is valid AND no single unused edge can be added. maximum is valid AND the collection size equals the global optimum over all valid matchings. An invalid collection receives neither certificate,even if no edge is addable or its numeric size equals the optimum. The exact result, retaining the requested order, is ([0, 1], No, [], No, 2, No, maximal_and_maximum_are_distinct_properties).
Reviewed step-by-step derivation
Maximum matching · qf-mod34-maxmatching-maximal-vs-maximum · candidate
81. Conjoin Validity Separately with Maximality and Maximum Size
Use the complete bounded simple graph on Fin(6)={0,…,5} with canonical ordered undirected edges=[[0, 1], [0, 2], [1, 2], [2, 3]]. Edge indices are zero-based positions in this exact list. A matching contains distinct edge indices with no shared endpoint. Enumerate every edge subset to determine the global maximum size; include the empty matching. List indices within each matching increasingly and enumerate matchings in increasing edge-subset mask order. Maximum,inclusion-maximal,perfect and unique are distinct properties. Proposed edge-index collection=[0, 1]. Give (matching,valid?,addable_edges,maximal?,maximum_size,maximum?,maximal_and_maximum_are_distinct_properties). List addable unselected edges increasingly when their endpoints avoid every endpoint used by the proposed collection.
View answer and explanation
([0, 1], No, [], No, 2, No, maximal_and_maximum_are_distinct_properties)
maximal is valid AND no single unused edge can be added. maximum is valid AND the collection size equals the global optimum over all valid matchings. An invalid collection receives neither certificate,even if no edge is addable or its numeric size equals the optimum. The exact result, retaining the requested order, is ([0, 1], No, [], No, 2, No, maximal_and_maximum_are_distinct_properties).
Reviewed step-by-step derivation
Maximum matching · qf-mod34-maxmatching-maximal-vs-maximum · candidate
82. Conjoin Validity Separately with Maximality and Maximum Size
Use the complete bounded simple graph on Fin(6)={0,…,5} with canonical ordered undirected edges=[[0, 1], [0, 3], [1, 2], [2, 3]]. Edge indices are zero-based positions in this exact list. A matching contains distinct edge indices with no shared endpoint. Enumerate every edge subset to determine the global maximum size; include the empty matching. List indices within each matching increasingly and enumerate matchings in increasing edge-subset mask order. Maximum,inclusion-maximal,perfect and unique are distinct properties. Proposed edge-index collection=[0, 1]. Give (matching,valid?,addable_edges,maximal?,maximum_size,maximum?,maximal_and_maximum_are_distinct_properties). List addable unselected edges increasingly when their endpoints avoid every endpoint used by the proposed collection.
View answer and explanation
([0, 1], No, [], No, 2, No, maximal_and_maximum_are_distinct_properties)
maximal is valid AND no single unused edge can be added. maximum is valid AND the collection size equals the global optimum over all valid matchings. An invalid collection receives neither certificate,even if no edge is addable or its numeric size equals the optimum. The exact result, retaining the requested order, is ([0, 1], No, [], No, 2, No, maximal_and_maximum_are_distinct_properties).
Reviewed step-by-step derivation
Maximum matching · qf-mod34-maxmatching-maximal-vs-maximum · candidate
83. Conjoin Validity Separately with Maximality and Maximum Size
Use the complete bounded simple graph on Fin(6)={0,…,5} with canonical ordered undirected edges=[[0, 1], [0, 2], [0, 3], [1, 2], [2, 3]]. Edge indices are zero-based positions in this exact list. A matching contains distinct edge indices with no shared endpoint. Enumerate every edge subset to determine the global maximum size; include the empty matching. List indices within each matching increasingly and enumerate matchings in increasing edge-subset mask order. Maximum,inclusion-maximal,perfect and unique are distinct properties. Proposed edge-index collection=[0, 1]. Give (matching,valid?,addable_edges,maximal?,maximum_size,maximum?,maximal_and_maximum_are_distinct_properties). List addable unselected edges increasingly when their endpoints avoid every endpoint used by the proposed collection.
View answer and explanation
([0, 1], No, [], No, 2, No, maximal_and_maximum_are_distinct_properties)
maximal is valid AND no single unused edge can be added. maximum is valid AND the collection size equals the global optimum over all valid matchings. An invalid collection receives neither certificate,even if no edge is addable or its numeric size equals the optimum. The exact result, retaining the requested order, is ([0, 1], No, [], No, 2, No, maximal_and_maximum_are_distinct_properties).
Reviewed step-by-step derivation
Maximum matching · qf-mod34-maxmatching-maximal-vs-maximum · candidate
84. Conjoin Validity Separately with Maximality and Maximum Size
Use the complete bounded simple graph on Fin(6)={0,…,5} with canonical ordered undirected edges=[[0, 1], [1, 3], [2, 3]]. Edge indices are zero-based positions in this exact list. A matching contains distinct edge indices with no shared endpoint. Enumerate every edge subset to determine the global maximum size; include the empty matching. List indices within each matching increasingly and enumerate matchings in increasing edge-subset mask order. Maximum,inclusion-maximal,perfect and unique are distinct properties. Proposed edge-index collection=[0, 1]. Give (matching,valid?,addable_edges,maximal?,maximum_size,maximum?,maximal_and_maximum_are_distinct_properties). List addable unselected edges increasingly when their endpoints avoid every endpoint used by the proposed collection.
View answer and explanation
([0, 1], No, [], No, 2, No, maximal_and_maximum_are_distinct_properties)
maximal is valid AND no single unused edge can be added. maximum is valid AND the collection size equals the global optimum over all valid matchings. An invalid collection receives neither certificate,even if no edge is addable or its numeric size equals the optimum. The exact result, retaining the requested order, is ([0, 1], No, [], No, 2, No, maximal_and_maximum_are_distinct_properties).
Reviewed step-by-step derivation
Maximum matching · qf-mod34-maxmatching-maximal-vs-maximum · candidate
85. Conjoin Validity Separately with Maximality and Maximum Size
Use the complete bounded simple graph on Fin(6)={0,…,5} with canonical ordered undirected edges=[[0, 1], [0, 2], [1, 3], [2, 3]]. Edge indices are zero-based positions in this exact list. A matching contains distinct edge indices with no shared endpoint. Enumerate every edge subset to determine the global maximum size; include the empty matching. List indices within each matching increasingly and enumerate matchings in increasing edge-subset mask order. Maximum,inclusion-maximal,perfect and unique are distinct properties. Proposed edge-index collection=[0, 1]. Give (matching,valid?,addable_edges,maximal?,maximum_size,maximum?,maximal_and_maximum_are_distinct_properties). List addable unselected edges increasingly when their endpoints avoid every endpoint used by the proposed collection.
View answer and explanation
([0, 1], No, [], No, 2, No, maximal_and_maximum_are_distinct_properties)
maximal is valid AND no single unused edge can be added. maximum is valid AND the collection size equals the global optimum over all valid matchings. An invalid collection receives neither certificate,even if no edge is addable or its numeric size equals the optimum. The exact result, retaining the requested order, is ([0, 1], No, [], No, 2, No, maximal_and_maximum_are_distinct_properties).
Reviewed step-by-step derivation
Maximum matching · qf-mod34-maxmatching-maximal-vs-maximum · candidate
86. Conjoin Validity Separately with Maximality and Maximum Size
Use the complete bounded simple graph on Fin(6)={0,…,5} with canonical ordered undirected edges=[[0, 1], [0, 3], [1, 3], [2, 3]]. Edge indices are zero-based positions in this exact list. A matching contains distinct edge indices with no shared endpoint. Enumerate every edge subset to determine the global maximum size; include the empty matching. List indices within each matching increasingly and enumerate matchings in increasing edge-subset mask order. Maximum,inclusion-maximal,perfect and unique are distinct properties. Proposed edge-index collection=[0, 1]. Give (matching,valid?,addable_edges,maximal?,maximum_size,maximum?,maximal_and_maximum_are_distinct_properties). List addable unselected edges increasingly when their endpoints avoid every endpoint used by the proposed collection.
View answer and explanation
([0, 1], No, [], No, 2, No, maximal_and_maximum_are_distinct_properties)
maximal is valid AND no single unused edge can be added. maximum is valid AND the collection size equals the global optimum over all valid matchings. An invalid collection receives neither certificate,even if no edge is addable or its numeric size equals the optimum. The exact result, retaining the requested order, is ([0, 1], No, [], No, 2, No, maximal_and_maximum_are_distinct_properties).
Reviewed step-by-step derivation
Maximum matching · qf-mod34-maxmatching-maximal-vs-maximum · candidate
87. Conjoin Validity Separately with Maximality and Maximum Size
Use the complete bounded simple graph on Fin(6)={0,…,5} with canonical ordered undirected edges=[[0, 1], [0, 2], [0, 3], [1, 3], [2, 3]]. Edge indices are zero-based positions in this exact list. A matching contains distinct edge indices with no shared endpoint. Enumerate every edge subset to determine the global maximum size; include the empty matching. List indices within each matching increasingly and enumerate matchings in increasing edge-subset mask order. Maximum,inclusion-maximal,perfect and unique are distinct properties. Proposed edge-index collection=[0, 1]. Give (matching,valid?,addable_edges,maximal?,maximum_size,maximum?,maximal_and_maximum_are_distinct_properties). List addable unselected edges increasingly when their endpoints avoid every endpoint used by the proposed collection.
View answer and explanation
([0, 1], No, [], No, 2, No, maximal_and_maximum_are_distinct_properties)
maximal is valid AND no single unused edge can be added. maximum is valid AND the collection size equals the global optimum over all valid matchings. An invalid collection receives neither certificate,even if no edge is addable or its numeric size equals the optimum. The exact result, retaining the requested order, is ([0, 1], No, [], No, 2, No, maximal_and_maximum_are_distinct_properties).
Reviewed step-by-step derivation
Maximum matching · qf-mod34-maxmatching-maximal-vs-maximum · candidate
88. Conjoin Validity Separately with Maximality and Maximum Size
Use the complete bounded simple graph on Fin(6)={0,…,5} with canonical ordered undirected edges=[[0, 1], [1, 2], [1, 3], [2, 3]]. Edge indices are zero-based positions in this exact list. A matching contains distinct edge indices with no shared endpoint. Enumerate every edge subset to determine the global maximum size; include the empty matching. List indices within each matching increasingly and enumerate matchings in increasing edge-subset mask order. Maximum,inclusion-maximal,perfect and unique are distinct properties. Proposed edge-index collection=[0, 1]. Give (matching,valid?,addable_edges,maximal?,maximum_size,maximum?,maximal_and_maximum_are_distinct_properties). List addable unselected edges increasingly when their endpoints avoid every endpoint used by the proposed collection.
View answer and explanation
([0, 1], No, [], No, 2, No, maximal_and_maximum_are_distinct_properties)
maximal is valid AND no single unused edge can be added. maximum is valid AND the collection size equals the global optimum over all valid matchings. An invalid collection receives neither certificate,even if no edge is addable or its numeric size equals the optimum. The exact result, retaining the requested order, is ([0, 1], No, [], No, 2, No, maximal_and_maximum_are_distinct_properties).
Reviewed step-by-step derivation
Maximum matching · qf-mod34-maxmatching-maximal-vs-maximum · candidate
89. Conjoin Validity Separately with Maximality and Maximum Size
Use the complete bounded simple graph on Fin(6)={0,…,5} with canonical ordered undirected edges=[[0, 1], [0, 2], [1, 2], [1, 3], [2, 3]]. Edge indices are zero-based positions in this exact list. A matching contains distinct edge indices with no shared endpoint. Enumerate every edge subset to determine the global maximum size; include the empty matching. List indices within each matching increasingly and enumerate matchings in increasing edge-subset mask order. Maximum,inclusion-maximal,perfect and unique are distinct properties. Proposed edge-index collection=[0, 1]. Give (matching,valid?,addable_edges,maximal?,maximum_size,maximum?,maximal_and_maximum_are_distinct_properties). List addable unselected edges increasingly when their endpoints avoid every endpoint used by the proposed collection.
View answer and explanation
([0, 1], No, [], No, 2, No, maximal_and_maximum_are_distinct_properties)
maximal is valid AND no single unused edge can be added. maximum is valid AND the collection size equals the global optimum over all valid matchings. An invalid collection receives neither certificate,even if no edge is addable or its numeric size equals the optimum. The exact result, retaining the requested order, is ([0, 1], No, [], No, 2, No, maximal_and_maximum_are_distinct_properties).
Reviewed step-by-step derivation
Maximum matching · qf-mod34-maxmatching-maximal-vs-maximum · candidate
90. Conjoin Validity Separately with Maximality and Maximum Size
Use the complete bounded simple graph on Fin(6)={0,…,5} with canonical ordered undirected edges=[[0, 1], [0, 3], [1, 2], [1, 3], [2, 3]]. Edge indices are zero-based positions in this exact list. A matching contains distinct edge indices with no shared endpoint. Enumerate every edge subset to determine the global maximum size; include the empty matching. List indices within each matching increasingly and enumerate matchings in increasing edge-subset mask order. Maximum,inclusion-maximal,perfect and unique are distinct properties. Proposed edge-index collection=[0, 1]. Give (matching,valid?,addable_edges,maximal?,maximum_size,maximum?,maximal_and_maximum_are_distinct_properties). List addable unselected edges increasingly when their endpoints avoid every endpoint used by the proposed collection.
View answer and explanation
([0, 1], No, [], No, 2, No, maximal_and_maximum_are_distinct_properties)
maximal is valid AND no single unused edge can be added. maximum is valid AND the collection size equals the global optimum over all valid matchings. An invalid collection receives neither certificate,even if no edge is addable or its numeric size equals the optimum. The exact result, retaining the requested order, is ([0, 1], No, [], No, 2, No, maximal_and_maximum_are_distinct_properties).
Reviewed step-by-step derivation
Maximum matching · qf-mod34-maxmatching-maximal-vs-maximum · candidate
91. Conjoin Validity Separately with Maximality and Maximum Size
Use the complete bounded simple graph on Fin(6)={0,…,5} with canonical ordered undirected edges=[[0, 1], [0, 2], [0, 3], [1, 2], [1, 3], [2, 3]]. Edge indices are zero-based positions in this exact list. A matching contains distinct edge indices with no shared endpoint. Enumerate every edge subset to determine the global maximum size; include the empty matching. List indices within each matching increasingly and enumerate matchings in increasing edge-subset mask order. Maximum,inclusion-maximal,perfect and unique are distinct properties. Proposed edge-index collection=[0, 1]. Give (matching,valid?,addable_edges,maximal?,maximum_size,maximum?,maximal_and_maximum_are_distinct_properties). List addable unselected edges increasingly when their endpoints avoid every endpoint used by the proposed collection.
View answer and explanation
([0, 1], No, [], No, 2, No, maximal_and_maximum_are_distinct_properties)
maximal is valid AND no single unused edge can be added. maximum is valid AND the collection size equals the global optimum over all valid matchings. An invalid collection receives neither certificate,even if no edge is addable or its numeric size equals the optimum. The exact result, retaining the requested order, is ([0, 1], No, [], No, 2, No, maximal_and_maximum_are_distinct_properties).
Reviewed step-by-step derivation
Maximum matching · qf-mod34-maxmatching-maximal-vs-maximum · candidate
92. Conjoin Validity Separately with Maximality and Maximum Size
Use the complete bounded simple graph on Fin(6)={0,…,5} with canonical ordered undirected edges=[[0, 1], [2, 3], [4, 5]]. Edge indices are zero-based positions in this exact list. A matching contains distinct edge indices with no shared endpoint. Enumerate every edge subset to determine the global maximum size; include the empty matching. List indices within each matching increasingly and enumerate matchings in increasing edge-subset mask order. Maximum,inclusion-maximal,perfect and unique are distinct properties. Proposed edge-index collection=[0, 1]. Give (matching,valid?,addable_edges,maximal?,maximum_size,maximum?,maximal_and_maximum_are_distinct_properties). List addable unselected edges increasingly when their endpoints avoid every endpoint used by the proposed collection.
View answer and explanation
([0, 1], Yes, [2], No, 3, No, maximal_and_maximum_are_distinct_properties)
maximal is valid AND no single unused edge can be added. maximum is valid AND the collection size equals the global optimum over all valid matchings. An invalid collection receives neither certificate,even if no edge is addable or its numeric size equals the optimum. The exact result, retaining the requested order, is ([0, 1], Yes, [2], No, 3, No, maximal_and_maximum_are_distinct_properties).
Reviewed step-by-step derivation
Maximum matching · qf-mod34-maxmatching-maximal-vs-maximum · candidate
93. Conjoin Validity Separately with Maximality and Maximum Size
Use the complete bounded simple graph on Fin(6)={0,…,5} with canonical ordered undirected edges=[[0, 1], [0, 2], [2, 3], [4, 5]]. Edge indices are zero-based positions in this exact list. A matching contains distinct edge indices with no shared endpoint. Enumerate every edge subset to determine the global maximum size; include the empty matching. List indices within each matching increasingly and enumerate matchings in increasing edge-subset mask order. Maximum,inclusion-maximal,perfect and unique are distinct properties. Proposed edge-index collection=[0, 1]. Give (matching,valid?,addable_edges,maximal?,maximum_size,maximum?,maximal_and_maximum_are_distinct_properties). List addable unselected edges increasingly when their endpoints avoid every endpoint used by the proposed collection.
View answer and explanation
([0, 1], No, [3], No, 3, No, maximal_and_maximum_are_distinct_properties)
maximal is valid AND no single unused edge can be added. maximum is valid AND the collection size equals the global optimum over all valid matchings. An invalid collection receives neither certificate,even if no edge is addable or its numeric size equals the optimum. The exact result, retaining the requested order, is ([0, 1], No, [3], No, 3, No, maximal_and_maximum_are_distinct_properties).
Reviewed step-by-step derivation
Maximum matching · qf-mod34-maxmatching-maximal-vs-maximum · candidate
94. Conjoin Validity Separately with Maximality and Maximum Size
Use the complete bounded simple graph on Fin(6)={0,…,5} with canonical ordered undirected edges=[[0, 1], [0, 3], [2, 3], [4, 5]]. Edge indices are zero-based positions in this exact list. A matching contains distinct edge indices with no shared endpoint. Enumerate every edge subset to determine the global maximum size; include the empty matching. List indices within each matching increasingly and enumerate matchings in increasing edge-subset mask order. Maximum,inclusion-maximal,perfect and unique are distinct properties. Proposed edge-index collection=[0, 1]. Give (matching,valid?,addable_edges,maximal?,maximum_size,maximum?,maximal_and_maximum_are_distinct_properties). List addable unselected edges increasingly when their endpoints avoid every endpoint used by the proposed collection.
View answer and explanation
([0, 1], No, [3], No, 3, No, maximal_and_maximum_are_distinct_properties)
maximal is valid AND no single unused edge can be added. maximum is valid AND the collection size equals the global optimum over all valid matchings. An invalid collection receives neither certificate,even if no edge is addable or its numeric size equals the optimum. The exact result, retaining the requested order, is ([0, 1], No, [3], No, 3, No, maximal_and_maximum_are_distinct_properties).
Reviewed step-by-step derivation
Maximum matching · qf-mod34-maxmatching-maximal-vs-maximum · candidate
95. Conjoin Validity Separately with Maximality and Maximum Size
Use the complete bounded simple graph on Fin(6)={0,…,5} with canonical ordered undirected edges=[[0, 1], [0, 2], [0, 3], [2, 3], [4, 5]]. Edge indices are zero-based positions in this exact list. A matching contains distinct edge indices with no shared endpoint. Enumerate every edge subset to determine the global maximum size; include the empty matching. List indices within each matching increasingly and enumerate matchings in increasing edge-subset mask order. Maximum,inclusion-maximal,perfect and unique are distinct properties. Proposed edge-index collection=[0, 1]. Give (matching,valid?,addable_edges,maximal?,maximum_size,maximum?,maximal_and_maximum_are_distinct_properties). List addable unselected edges increasingly when their endpoints avoid every endpoint used by the proposed collection.
View answer and explanation
([0, 1], No, [4], No, 3, No, maximal_and_maximum_are_distinct_properties)
maximal is valid AND no single unused edge can be added. maximum is valid AND the collection size equals the global optimum over all valid matchings. An invalid collection receives neither certificate,even if no edge is addable or its numeric size equals the optimum. The exact result, retaining the requested order, is ([0, 1], No, [4], No, 3, No, maximal_and_maximum_are_distinct_properties).
Reviewed step-by-step derivation
Maximum matching · qf-mod34-maxmatching-maximal-vs-maximum · candidate
96. Conjoin Validity Separately with Maximality and Maximum Size
Use the complete bounded simple graph on Fin(6)={0,…,5} with canonical ordered undirected edges=[[0, 1], [1, 2], [2, 3], [4, 5]]. Edge indices are zero-based positions in this exact list. A matching contains distinct edge indices with no shared endpoint. Enumerate every edge subset to determine the global maximum size; include the empty matching. List indices within each matching increasingly and enumerate matchings in increasing edge-subset mask order. Maximum,inclusion-maximal,perfect and unique are distinct properties. Proposed edge-index collection=[0, 1]. Give (matching,valid?,addable_edges,maximal?,maximum_size,maximum?,maximal_and_maximum_are_distinct_properties). List addable unselected edges increasingly when their endpoints avoid every endpoint used by the proposed collection.
View answer and explanation
([0, 1], No, [3], No, 3, No, maximal_and_maximum_are_distinct_properties)
maximal is valid AND no single unused edge can be added. maximum is valid AND the collection size equals the global optimum over all valid matchings. An invalid collection receives neither certificate,even if no edge is addable or its numeric size equals the optimum. The exact result, retaining the requested order, is ([0, 1], No, [3], No, 3, No, maximal_and_maximum_are_distinct_properties).
Reviewed step-by-step derivation
Maximum matching · qf-mod34-maxmatching-maximal-vs-maximum · candidate
97. Conjoin Validity Separately with Maximality and Maximum Size
Use the complete bounded simple graph on Fin(6)={0,…,5} with canonical ordered undirected edges=[[0, 1], [0, 2], [1, 2], [2, 3], [4, 5]]. Edge indices are zero-based positions in this exact list. A matching contains distinct edge indices with no shared endpoint. Enumerate every edge subset to determine the global maximum size; include the empty matching. List indices within each matching increasingly and enumerate matchings in increasing edge-subset mask order. Maximum,inclusion-maximal,perfect and unique are distinct properties. Proposed edge-index collection=[0, 1]. Give (matching,valid?,addable_edges,maximal?,maximum_size,maximum?,maximal_and_maximum_are_distinct_properties). List addable unselected edges increasingly when their endpoints avoid every endpoint used by the proposed collection.
View answer and explanation
([0, 1], No, [4], No, 3, No, maximal_and_maximum_are_distinct_properties)
maximal is valid AND no single unused edge can be added. maximum is valid AND the collection size equals the global optimum over all valid matchings. An invalid collection receives neither certificate,even if no edge is addable or its numeric size equals the optimum. The exact result, retaining the requested order, is ([0, 1], No, [4], No, 3, No, maximal_and_maximum_are_distinct_properties).
Reviewed step-by-step derivation
Maximum matching · qf-mod34-maxmatching-maximal-vs-maximum · candidate
98. Conjoin Validity Separately with Maximality and Maximum Size
Use the complete bounded simple graph on Fin(6)={0,…,5} with canonical ordered undirected edges=[[0, 1], [0, 3], [1, 2], [2, 3], [4, 5]]. Edge indices are zero-based positions in this exact list. A matching contains distinct edge indices with no shared endpoint. Enumerate every edge subset to determine the global maximum size; include the empty matching. List indices within each matching increasingly and enumerate matchings in increasing edge-subset mask order. Maximum,inclusion-maximal,perfect and unique are distinct properties. Proposed edge-index collection=[0, 1]. Give (matching,valid?,addable_edges,maximal?,maximum_size,maximum?,maximal_and_maximum_are_distinct_properties). List addable unselected edges increasingly when their endpoints avoid every endpoint used by the proposed collection.
View answer and explanation
([0, 1], No, [4], No, 3, No, maximal_and_maximum_are_distinct_properties)
maximal is valid AND no single unused edge can be added. maximum is valid AND the collection size equals the global optimum over all valid matchings. An invalid collection receives neither certificate,even if no edge is addable or its numeric size equals the optimum. The exact result, retaining the requested order, is ([0, 1], No, [4], No, 3, No, maximal_and_maximum_are_distinct_properties).
Reviewed step-by-step derivation
Maximum matching · qf-mod34-maxmatching-maximal-vs-maximum · candidate
99. Conjoin Validity Separately with Maximality and Maximum Size
Use the complete bounded simple graph on Fin(6)={0,…,5} with canonical ordered undirected edges=[[0, 1], [0, 2], [0, 3], [1, 2], [2, 3], [4, 5]]. Edge indices are zero-based positions in this exact list. A matching contains distinct edge indices with no shared endpoint. Enumerate every edge subset to determine the global maximum size; include the empty matching. List indices within each matching increasingly and enumerate matchings in increasing edge-subset mask order. Maximum,inclusion-maximal,perfect and unique are distinct properties. Proposed edge-index collection=[0, 1]. Give (matching,valid?,addable_edges,maximal?,maximum_size,maximum?,maximal_and_maximum_are_distinct_properties). List addable unselected edges increasingly when their endpoints avoid every endpoint used by the proposed collection.
View answer and explanation
([0, 1], No, [5], No, 3, No, maximal_and_maximum_are_distinct_properties)
maximal is valid AND no single unused edge can be added. maximum is valid AND the collection size equals the global optimum over all valid matchings. An invalid collection receives neither certificate,even if no edge is addable or its numeric size equals the optimum. The exact result, retaining the requested order, is ([0, 1], No, [5], No, 3, No, maximal_and_maximum_are_distinct_properties).
Reviewed step-by-step derivation
Maximum matching · qf-mod34-maxmatching-maximal-vs-maximum · candidate
100. Conjoin Validity Separately with Maximality and Maximum Size
Use the complete bounded simple graph on Fin(6)={0,…,5} with canonical ordered undirected edges=[[0, 1], [1, 3], [2, 3], [4, 5]]. Edge indices are zero-based positions in this exact list. A matching contains distinct edge indices with no shared endpoint. Enumerate every edge subset to determine the global maximum size; include the empty matching. List indices within each matching increasingly and enumerate matchings in increasing edge-subset mask order. Maximum,inclusion-maximal,perfect and unique are distinct properties. Proposed edge-index collection=[0, 1]. Give (matching,valid?,addable_edges,maximal?,maximum_size,maximum?,maximal_and_maximum_are_distinct_properties). List addable unselected edges increasingly when their endpoints avoid every endpoint used by the proposed collection.
View answer and explanation
([0, 1], No, [3], No, 3, No, maximal_and_maximum_are_distinct_properties)
maximal is valid AND no single unused edge can be added. maximum is valid AND the collection size equals the global optimum over all valid matchings. An invalid collection receives neither certificate,even if no edge is addable or its numeric size equals the optimum. The exact result, retaining the requested order, is ([0, 1], No, [3], No, 3, No, maximal_and_maximum_are_distinct_properties).
Reviewed step-by-step derivation
Maximum matching · qf-mod34-maxmatching-maximal-vs-maximum · candidate
Canonical question JSONL · English JSONL · Validation report