2-connected equimatchable graphs on surfaces

Abstract

A graph $G$ is equimatchable if any matching in $G$ is a subset of a maximum-size matching. It is known that any $2$-connected equimatchable graph is either bipartite or factor-critical. We prove that for any vertex $v$ of a $2$-connected factor-critical equimatchable graph $G$ and a minimal matching $M$ that isolates $v$ the graph $G\setminus(M\cup\{ v\})$ is either $K_{2n}$ or $K_{n,n}$ for some $n$. We use this result to improve the upper bounds on the maximum size of $2$-connected equimatchable factor-critical graphs embeddable in the orientable surface of genus $g$ to $4\sqrt g+17$ if $g\le 2$ and to $12\sqrt g+5$ if $g\ge 3$. Moreover, for any nonnegative integer $g$ we construct a $2$-connected equimatchable factor-critical graph with genus $g$ and more than $4\sqrt{2g}$ vertices, which establishes that the maximum size of such graphs is $\Theta(\sqrt g)$. Similar bounds are obtained also for nonorientable surfaces. Finally, 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 $k$.