Maximum δ‐edge‐colorable subgraphs of class II graphs

A graph Gis class II, if its chromatic index is at least δ + 1. Let Hbe 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