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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 ...
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 ...
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.
A complete bipartite graph K m,n has a maximum matching of size min{m,n}. A complete bipartite graph K n,n has a proper n-edge-coloring corresponding to a Latin square. [14] Every complete bipartite graph is a modular graph: every triple of vertices has a median that belongs to shortest paths between each pair of vertices. [15]
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} .
For example, consider the graph K 2,2: the complete bipartite graph on 2+2 vertices. Suppose the edges (x 1,y 1) and (x 2,y 2) are colored green, and the edges (x 1,y 2) and (x 2,y 1) are colored blue. This is a proper coloring, but there are only two perfect matchings, and each of them is colored by a single color.
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).
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").
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