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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 ...
The sum of weighted perfect matchings can also be computed by using the Tutte matrix for the adjacency matrix in the last step. Kuratowski's theorem states that a finite graph is planar if and only if it contains no subgraph homeomorphic to K 5 (complete graph on five vertices) or K 3,3 (complete bipartite graph on two partitions of size three).
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.
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]
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 ...
When modelling relations between two different classes of objects, bipartite graphs very often arise naturally. For instance, a graph of football players and clubs, with an edge between a player and a club if the player has played for that club, is a natural example of an affiliation network, a type of bipartite graph used in social network analysis.
Given two sets, A and T, together with a weight function C : A × T → R. Find a bijection f : A → T such that the cost function: (, ()) is minimized. Usually the weight function is viewed as a square real-valued matrix C, so that the cost function is written down as:
The hafnian of an adjacency matrix of a graph is the number of perfect matchings (also known as 1-factors) in the graph. This is because a partition of { 1 , 2 , … , 2 n } {\displaystyle \{1,2,\dots ,2n\}} into subsets of size 2 can also be thought of as a perfect matching in the complete graph K 2 n {\displaystyle K_{2n}} .