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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. 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 ...

  4. 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.

  5. 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).

  6. Edmonds matrix - Wikipedia

    en.wikipedia.org/wiki/Edmonds_matrix

    One application of the Edmonds matrix of a bipartite graph is that the graph admits a perfect matching if and only if the polynomial det(A ij) in the x ij is not identically zero. Furthermore, the number of perfect matchings is equal to the number of monomials in the polynomial det( A ), and is also equal to the permanent of A {\displaystyle A} .

  7. Dulmage–Mendelsohn decomposition - Wikipedia

    en.wikipedia.org/wiki/Dulmage–Mendelsohn...

    The Dulmage-Mendelshon decomposition can be constructed as follows. [2] (it is attributed to [3] who in turn attribute it to [4]).Let G be a bipartite graph, M a maximum-cardinality matching in G, and V 0 the set of vertices of G unmatched by M (the "free vertices").

  8. Hall-type theorems for hypergraphs - Wikipedia

    en.wikipedia.org/wiki/Hall-type_theorems_for...

    Any tripartite hypergraph (X 1 + X 2 + Y, E) in which | Y | = 2n – 1, the degree of each vertex y in Y is n, and the neighbor-set of y is a matching, has a matching of size n. [23] The 2n – 1 is the best possible: if | Y | = 2n – 2, then the maximum matching may be of size n-1.

  9. ♯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]