# 4.6: Matchings


A system of distinct representatives corresponds to a set of edges in the corresponding bipartite graph that share no endpoints; such a collection of edges (in any graph, not just a bipartite graph) is called a matching. In Figure $$\PageIndex{1}$$, a matching is shown in red. This is a largest possible matching, since it contains edges incident with all four of the top vertices, and it thus corresponds to a complete sdr.
Any bipartite graph can be interpreted as a set system: we simply label all the vertices in one part with "set names'' $$A_1$$, $$A_2$$, etc., and the other part is labeled with "element names'', and the sets are defined in the obvious way: $$A_i$$ is the neighborhood of the vertex labeled "$$A_i$$''. Thus, we know one way to compute the size of a maximum matching, namely, we interpret the bipartite graph as a set system and compute the size of a maximum sdr; this is the size of a maximum matching.