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    "title": "Edmonds: finding the largest pairing in a general graph",
    "summary": "The1965paper gives a conceptual polynomial-time method for maximum-cardinality matching in finite general graphs. Alternating trees and conditionally safe blossom contraction deal with odd cycles, with expansion rules that lift augmentations to the original graph.\n\nSuppose every edge represents an allowed pair and each participant may belong to at most one pair. In a general network, odd cycles can trap a simple search. Edmonds temporarily replaces an appropriate odd-cycle structure by one vertex, continues the search, and then expands it to recover a valid larger pairing. The paper also produces a matching/odd-set-cover duality certificate. It distinguishes proving an efficient algorithm exists from supplying complete production code.",
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        "description": "Finite general graph, maximum cardinality rather than maximum weight or hypergraph matching. Original 450–452 gives a conceptual algorithm and n^4 time/n^2 memory upper bounds under its step/storage accounting; no modern tightbound or measured runtime is claimed. Safe maximality preservation uses a blossom belonging to a flower/stem; arbitrary odd-cycle contraction is not a universal equivalence. Retained pseudovertices may need expansion when inner vertices, as 465–466 explain. The odd-set-cover duality is proved here; the stronger full matching-polytope/weighted optimization proof is explicitly deferred to another paper and excluded from this target. Original discussion of polynomial efficiency is informal foundational reasoning, not a formal proof that every polynomial algorithm is practical or a solution of Pversus NP. no independent proof reconstruction, runnable implementation or benchmark certification. Berge, Norman/Rabin, Tutte, network-flow predecessors, Dantzig discussions and Witzgall’s proof suggestion remain distinct inputs.",
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      "statement": "# Edmonds: finding the largest pairing in a general graph\n\nThe1965paper gives a conceptual polynomial-time method for maximum-cardinality matching in finite general graphs. Alternating trees and conditionally safe blossom contraction deal with odd cycles, with expansion rules that lift augmentations to the original graph.\n\nSuppose every edge represents an allowed pair and each participant may belong to at most one pair. In a general network, odd cycles can trap a simple search. Edmonds temporarily replaces an appropriate odd-cycle structure by one vertex, continues the search, and then expands it to recover a valid larger pairing. The paper also produces a matching/odd-set-cover duality certificate. It distinguishes proving an efficient algorithm exists from supplying complete production code.\n\n## Scientific scope\n\nFinite general graph, maximum cardinality rather than maximum weight or hypergraph matching. Original 450–452 gives a conceptual algorithm and n^4 time/n^2 memory upper bounds under its step/storage accounting; no modern tightbound or measured runtime is claimed. Safe maximality preservation uses a blossom belonging to a flower/stem; arbitrary odd-cycle contraction is not a universal equivalence. Retained pseudovertices may need expansion when inner vertices, as 465–466 explain. The odd-set-cover duality is proved here; the stronger full matching-polytope/weighted optimization proof is explicitly deferred to another paper and excluded from this target. Original discussion of polynomial efficiency is informal foundational reasoning, not a formal proof that every polynomial algorithm is practical or a solution of Pversus NP. no independent proof reconstruction, runnable implementation or benchmark certification. Berge, Norman/Rabin, Tutte, network-flow predecessors, Dantzig discussions and Witzgall’s proof suggestion remain distinct inputs.\n\n## People and systems\n\nJack Edmonds\n\n\n\n## Original sources\n\n- [Edmonds:Paths,Trees,and Flowers(1965), original449–467](https://www.cambridge.org/core/services/aop-cambridge-core/content/view/08B492B72322C4130AE800C0610E0E21/S0008414X00039419a.pdf/paths_trees_and_flowers.pdf)\n- [Cambridge:Paths,Trees,and Flowers original1965publication metadata](https://www.cambridge.org/core/journals/canadian-journal-of-mathematics/article/paths-trees-and-flowers/08B492B72322C4130AE800C0610E0E21)\n",
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