Maximum Matchings in Regular Graphs of High Girth

Let G = (V,E) be any d-regular graph with girth g on n vertices, for d ≥ 3. This note shows that G has a maximum matching which includes all but an exponentially small fraction of the vertices, O((d − 1)−g/2). Specifically, in a maximum matching of G, the number of unmatched vertices is at most n/n0(d, g), where n0(d, g) is the number of vertices in a ball of radius b(g − 1)/2c around a vertex, for odd values of g, and around an edge, for even values of g. This result is tight if n < 2n0(d, g).