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

TL;DR

This paper investigates the properties of maximum $ ext{Delta}$-edge-colorable subgraphs in class II graphs, establishing bounds and structural characteristics, and explores extensions of cycle sets and properties of simple graphs.

Contribution

It provides the best possible lower bounds for the ratio of edges in maximum $ ext{Delta}$-edge-colorable subgraphs and reveals structural properties and extensions in class II graphs.

Findings

Established lower bounds for edge ratios in subgraphs.

Showed that cycle sets can be extended to maximum subgraphs.

Proved that maximum subgraphs in simple graphs are class I.

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 $\Delta\geq3$ 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.