Lectures on Nonbipartite Matchings

Our first topic of study is matchings in graphs which are not necessarily bipartite. We begin with some relevant terminology and definitions. A matching is a set of edges that share no endvertices. A vertex v is covered by a matching if v is incident with an edge in the matching. A matching that covers every vertex is known as a perfect matching or a 1-factor (i.e., a spanning regular subgraph in which every vertex has degree 1). We will let ν(G) denote the cardinality of a maximum matching in graph G. A vertex cover is a set C of vertices such that every edge is incident with at least one vertex in C. The minimum cardinality of a vertex cover is denoted τ(G). The following simple proposition relates matchings and vertex covers.