Equimatchable Graphs on Surfaces

Abstract

A graph G is equimatchable if each matching in G is a subset of a maximum-size matching and it is factor critical if G - v has a perfect matching for each vertex v of G. It is known that any 2-connected equimatchable graph is either bipartite or factor critical. We prove that for 2-connected factor-critical equimatchable graph G the graph G \ (V(M) ∪ {v}) is either K _2n or K _n,n for some n for any vertex v of G and any minimal matching M such that {v} is a component of G \ V(M). We use this result to improve the upper bounds on the maximum number of vertices of 2-connected equimatchable factor-critical graphs embeddable in the orientable surface of genus g to 4√g + 17 if g ≤ 2 and to 12√g + 5 if g ≥ 3. Moreover, for any nonnegative integer g we construct a 2-connected equimatchable factor-critical graph with genus g and more than 4√2g vertices, which establishes that the maximum size of such graphs is Θ(√g). Similar bounds are obtained also for nonorientable surfaces. In the bipartite case for any nonnegative integers g, h, and k we provide a construction of arbitrarily large 2-connected equimatchable bipartite graphs with orientable genus g, respectively nonorientable genus h, and a genus embedding with face-width k. Finally, we prove that any d-degenerate 2-connected equimatchable factor-critical graph has at most 4d + 1 vertices, where a graph is d-degenerate if every its induced subgraph contains a vertex of degree at most d.