Uniquely restricted matchings and edge colorings

Abstract: A matching in a graph is uniquely restricted if no other matching covers exactly the same set of vertices. This notion was defined by Golumbic, Hirst, and Lewenstein and studied in a number of articles. Our contribution is twofold. We provide approximation algorithms for computing a uniquely restricted matching of maximum size in some bipartite graphs. In particular, we achieve a ratio of $9/5$ for subcubic bipartite graphs, improving over a $2$-approximation algorithm proposed by Mishra. Furthermore, we study the uniquely restricted chromatic index of a graph, defined as the minimum number of uniquely restricted matchings into which its edge set can be partitioned. We provide tight upper bounds in terms of the maximum degree and characterize all extremal graphs. Our constructive proofs yield efficient algorithms to determine the corresponding edge colorings.