Degenerate matchings and edge colorings

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.