Maximum $\Delta$-edge-colorable subgraphs of class II graphs

Abstract

A graph $G$ is class II, if its chromatic index is at least $\Delta +1$. Let $H$ be a maximum $\Delta$-edge-colorable subgraph of $G$. The paper proves best possible lower bounds for $\frac{|E(H)|}{|E(G)|}$, and structural properties of maximum $\Delta$-edge-colorable subgraphs. It is shown that every set of vertex-disjoint cycles of a class II graph with $\Delta \ge 3$ can be extended to a maximum $\Delta$-edge-colorable subgraph. Simple graphs have a maximum $\Delta$-edge-colorable subgraph such that the complement is a matching. Furthermore, a maximum $\Delta$-edge-colorable subgraph of a simple graph is always class I.

13 pages, 2 figures, the proof of the Lemma 1 is corrected

References in corpus (2)

• On disjoint matchings in cubic graphs
• On disjoint matchings in cubic graphs: maximum 2- and 3-edge-colorable subgraphs

Cited by in corpus (2)

• Maximal $k$-Edge-Colorable Subgraphs, Vizing's Theorem, and Tuza's Conjecture
• Measures of edge-uncolorability