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  2. Matching (graph theory) - Wikipedia

    en.wikipedia.org/wiki/Matching_(graph_theory)

    The number of perfect matchings in a complete graph K n (with n even) is given by the double factorial (n − 1)!!. [13] The numbers of matchings in complete graphs, without constraining the matchings to be perfect, are given by the telephone numbers. [14] The number of perfect matchings in a graph is also known as the hafnian of its adjacency ...

  3. Perfect matching - Wikipedia

    en.wikipedia.org/wiki/Perfect_matching

    However, counting the number of perfect matchings, even in bipartite graphs, is #P-complete. This is because computing the permanent of an arbitrary 0–1 matrix (another #P-complete problem) is the same as computing the number of perfect matchings in the bipartite graph having the given matrix as its biadjacency matrix.

  4. Maximum cardinality matching - Wikipedia

    en.wikipedia.org/wiki/Maximum_cardinality_matching

    For sparse bipartite graphs, the maximum matching problem can be solved in ~ (/) with Madry's algorithm based on electric flows. [3] For planar bipartite graphs, the problem can be solved in time O(n log 3 n) where n is the number of vertices, by reducing the problem to maximum flow with multiple sources and sinks. [4]

  5. Kőnig's theorem (graph theory) - Wikipedia

    en.wikipedia.org/wiki/Kőnig's_theorem_(graph...

    Kőnig had announced in 1914 and published in 1916 the results that every regular bipartite graph has a perfect matching, [11] and more generally that the chromatic index of any bipartite graph (that is, the minimum number of matchings into which it can be partitioned) equals its maximum degree [12] – the latter statement is known as Kőnig's ...

  6. Petersen's theorem - Wikipedia

    en.wikipedia.org/wiki/Petersen's_theorem

    It was conjectured by Lovász and Plummer that the number of perfect matchings contained in a cubic, bridgeless graph is exponential in the number of the vertices of the graph n. [5] The conjecture was first proven for bipartite, cubic, bridgeless graphs by Voorhoeve (1979), later for planar, cubic, bridgeless graphs by Chudnovsky & Seymour (2012).

  7. Complete graph - Wikipedia

    en.wikipedia.org/wiki/Complete_graph

    The number of perfect matchings of the complete graph K n (with n even) is given by the double factorial (n – 1)!!. [12] The crossing numbers up to K 27 are known, with K 28 requiring either 7233 or 7234 crossings. Further values are collected by the Rectilinear Crossing Number project. [13] Rectilinear Crossing numbers for K n are

  8. ♯P-completeness of 01-permanent - Wikipedia

    en.wikipedia.org/wiki/%E2%99%AFP-completeness_of...

    Since any 0–1 matrix is the biadjacency matrix of some bipartite graph, Valiant's theorem implies [9] that the problem of counting the number of perfect matchings in a bipartite graph is #P-complete, and in conjunction with Toda's theorem this implies that it is hard for the entire polynomial hierarchy. [10] [11]

  9. Matching in hypergraphs - Wikipedia

    en.wikipedia.org/wiki/Matching_in_hypergraphs

    The matching number of a hypergraph H is the largest size of a ... Then H admits several matchings of size 2, for example: { {1,2,3}, {4,5,6} } ... in a bipartite ...