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Unformatted text preview: MATH 682 Notes Combinatorics and Graph Theory II 1 Matchings A popular question to be asked on graphs, if graphs represent some sort of compatability or asso- ciation, is how to associate as many vertices as possible into well-matched pairs. It is to this end that we introduce the concept of a matching : Definition 1. A matching on a graph G is a subset M of the edge-set E ( G ) such that every vertex of G is incident on at most one edge of M . A matching can be thought of as representing a pairing of vertices in which only adjacent vertices are paired, and each vertex is paired with only one of its neighbors. A matching might not include every vertex of the graph, but in investigating the question of how many vertices we can match with good partners, doing so is obviously a desirable effect. We thus introduce two significant concepts with regard to a matchs quality: Definition 2. A matching M is maximal in a graph G if there is no matching M with | M | > | M | . Definition 3. A mathing M is a perfect matching , also called a 1-factor , if the edges of M are incident to every vertex of G . The term 1-factor is a bit mysterious, so, although we will not investigate it further at this time, we present the general concept of which a matching is a special case. Definition 4. A subgraph H of G is called a k-factor if V ( H ) = V ( G ) and H is k-regular. Since 1-regular graphs are graphs consisting of unions of non-incident edges, a matching corresponds exactly to a 1-regular subgraph, and a perfect matching to a 1-factor. We may make a few very simple related observations: The edges of a matching M are incident on 2 | M | vertices. A matching is perfect in G if and only if | M | = 1 2 | G | . If a graph has perfect matchings, the set of perfect matchings is identical to the set of maximal matchings. G has no perfect matchings if | G | is odd. 1.1 Matchings in bipartite graphs The question of finding maximal matchings on arbitrary graphs is sufficiently complicated that at first we should constrain ourselves to a simple but useful special case: when G is bipartite. Here the graph represents a common real-world problem: we have sets A and B of unlike objects, a particular set of pairs elements of A and B which we are allowed to associate with eachother, and a goal of pairing off as many of these elements as possible. Traditionally this problem has often been labeled the marriage problem after the traditional context of arranging men and women, not all of whom are necessarily compatible with each other, into appropriate marriages (note: in several U.S. states marriage arrangement is no longer guaranteed to be bipartite, but the appellation stands)....
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