Measures of edge-uncolorability

TL;DR

This paper investigates the resistance of graphs to becoming edge-colorable with maximum degree edges, relating it to structural properties and vertex removal parameters, providing bounds and optimality results.

Contribution

It introduces bounds relating graph resistance to vertex removal measures and explores these concepts in regular and general graphs, establishing optimal bounds.

Findings

The ratio of resistance to vertex removal measure is at most half the maximum degree.

The established bound is proven to be tight and optimal.

The paper connects resistance with structural properties like 2-factors.

Abstract

The resistance $r(G)$ of a graph $G$ is the minimum number of edges that have to be removed from $G$ to obtain a graph which is $\Delta(G)$-edge-colorable. The paper relates the resistance to other parameters that measure how far is a graph from being $\Delta$-edge-colorable. The first part considers regular graphs and the relation of the resistance to structural properties in terms of 2-factors. The second part studies general (multi-) graphs $G$. Let $r_v(G)$ be the minimum number of vertices that have to be removed from $G$ to obtain a class 1 graph. We show that $\frac{r(G)}{r_v(G)} \leq \lfloor \frac{\Delta(G)}{2} \rfloor$, and that this bound is best possible.