Maximum Δ-edge-colorable subgraphs of class II graphs

A graph G is class II, if its chromatic index is at least δ + 1. Let H be a maximum δ-edge-colorable subgraph of G. The paper proves best possible lower bounds for |E(H)|/|E(G)|, and structural properties of maximum δ-edge-colorable subgraphs. It is shown that every set of vertex-disjoint cycles of a class II graph with δ≥3 can be extended to a maximum δ-edge-colorable subgraph. Simple graphs have a maximum δ-edge-colorable subgraph such that the complement is a matching. Furthermore, a maximum δ-edge-colorable subgraph of a simple graph is always class I. © 2011 Wiley Periodicals, Inc. J Graph Theory © 2012 Wiley Periodicals, Inc.