Uniquely restricted matchings in subcubic graphs without short cycles

Abstract

A matching $M$ in a graph $G$ is uniquely restricted if no other matching in $G$ covers the same set of vertices. We prove that any connected subcubic graph with $n$ vertices and girth at least $5$ contains a uniquely restricted matching of size at least $(n-1)/3$ except for two exceptional cubic graphs of order $14$ and $20$.