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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 ...
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]
Equivalently it asks for the maximum number of edges in a balanced bipartite graph whose edges can be partitioned into a linear number of induced matchings, or the maximum number of triples one can choose from points so that every six points contain at most two triples.
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 ...
The number of possible perfect matchings in a bipartite graph with equal-sized partitions can therefore be estimated via the degrees of the vertices of any of the two partitions. [ 7 ] Related statements
For every subset F of edges, the dot product 1 E(v) · 1 F represents the number of edges in F that are adjacent to v. Therefore, the following statements are equivalent: A subset F of edges represents a matching in G; For every node v in V: 1 E(v) · 1 F ≤ 1. A G · 1 F ≤ 1 V. The cardinality of a set F of edges is the dot product 1 E · 1 F.
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 .
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.