On Graphs with a Unique Perfect Matching

Abstract

If a graph has a unique perfect matching, we call it a UPM-graph. In this paper we study UPM-graphs. It was shown by Kotzig that a connected UPM-graph has a cut edge belonging to its unique perfect matching. We strengthen this result to a further structural characterization. Using the stronger result, we present a characterization of claw-free UPM-graphs, and prove that for any fixed positive integer $$n$$n, the number of edges of saturated UPM-graphs on $$2n$$2n vertices form an arithmetic progression from $$(2n+2)\lfloor \log _2(n+1)\rfloor -2^{2+\lfloor \log _2(n+1)\rfloor }+n+4$$(2n+2)?log2(n+1)?-22+?log2(n+1)?+n+4 to $$n^2$$n2 with common difference 2. For a fixed positive integer $$n$$n, we determine the number of labelled UPM-trees on $$2n$$2n vertices. For a bipartite UPM-graph which has maximum number of edges, we determine the number of spanning UPM-trees of it.