Degenerate Matchings and Edge Colorings

TL;DR

This paper introduces efficient algorithms for finding maximum $r$-degenerate matchings in chordal graphs and explores the $r$-chromatic index, providing bounds and extremal graph characterizations.

Contribution

It presents an efficient algorithm for maximum $r$-degenerate matchings in chordal graphs and studies the $r$-chromatic index with bounds and extremal cases.

Findings

Efficient algorithm for maximum $r$-degenerate matching in chordal graphs.

Upper bounds for the $r$-chromatic index.

Characterization of extremal graphs for the $r$-chromatic index.

Abstract

A matching $M$ in a graph $G$ is $r$-degenerate if the subgraph of $G$ induced by the set of vertices incident with an edge in $M$ is $r$-degenerate. Goddard, Hedetniemi, Hedetniemi, and Laskar (Generalized subgraph-restricted matchings in graphs, Discrete Mathematics 293 (2005) 129-138) introduced the notion of acyclic matchings, which coincide with $1$-degenerate matchings. Solving a problem they posed, we describe an efficient algorithm to determine the maximum size of an $r$-degenerate matching in a given chordal graph. Furthermore, we study the $r$-chromatic index of a graph defined as the minimum number of $r$-degenerate matchings into which its edge set can be partitioned, obtaining upper bounds and discussing extremal graphs.