Uniquely restricted matchings in subcubic graphs

A matching M in a graph G is uniquely restricted if no other matching in G covers the same set of vertices. We conjecture that every connected subcubic graph with m edges and b bridges that is distinct from K _3,3 has a uniquely restricted matching of size at least m/+b6, and we establish this bound with b replaced by the number of bridges that lie on a path between two vertices of degree at most 2. Moreover, we prove that every connected subcubic graph of order n and girth at least 7 has a uniquely restricted matching of size at least n/−13, which partially confirms a conjecture of Fürst and Rautenbach (Graphs Combin. 35 (2019) 353–361).