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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 ...
Use the planar embedding to create an (undirected) graph T 2 with the same vertex set as the dual graph of G. Create an edge in T 2 between two vertices if their corresponding faces in G share an edge in G that is not in T 1. (Note that T 2 is a tree.) For each leaf v in T 2 (that is not also the root):
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 ...
[1] [2] Further entropy-based proofs have been given by Alexander Schrijver and Jaikumar Radhakrishnan. [ 3 ] [ 4 ] The Bregman–Minc inequality is used, for example, in graph theory to obtain upper bounds for the number of perfect matchings in a bipartite graph .
To convert the bipartite graph induced matching problem into the unique triangle problem, add a third set of vertices to the graph, one for each induced matching, and add edges from vertices and of the bipartite graph to vertex in this third set whenever bipartite edge belongs to induced 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.
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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]
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