Uniquely restricted matchings and edge colorings

TL;DR

This paper introduces approximation algorithms for maximum uniquely restricted matchings in bipartite graphs and explores the uniquely restricted chromatic index, providing bounds, characterizations, and efficient coloring algorithms.

Contribution

It offers improved approximation algorithms for maximum uniquely restricted matchings and characterizes the uniquely restricted chromatic index with tight bounds and constructive methods.

Findings

Achieved a 9/5 approximation ratio for subcubic bipartite graphs.

Provided tight upper bounds for the uniquely restricted chromatic index.

Developed efficient algorithms for edge coloring based on uniquely restricted matchings.

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.