Overfullness of critical class 2 graphs with a small core degree

TL;DR

This paper proves that certain critical class 2 graphs with a small core degree are overfull, confirming a conjecture relating overfullness to the core structure and maximum degree.

Contribution

It extends Hilton and Zhao's conjecture by showing that if the core's minimum degree is at most two and the maximum degree exceeds half the order plus one, then the graph is overfull.

Findings

Critical class 2 graphs with small core degree are overfull under specified conditions.

The result confirms a conjecture relating core structure and overfullness.

Provides new insights into the structure of graphs with high chromatic index.

Abstract

Let $G$ be a simple graph, and let $n$, $\Delta(G)$ and $\chi' (G)$ be the order, the maximum degree and the chromatic index of $G$, respectively. We call $G$ overfull if $|E(G)|/\lfloor n/2\rfloor > \Delta(G)$, and critical if $\chi'(H) < \chi'(G)$ for every proper subgraph $H$ of $G$. Clearly, if $G$ is overfull then $\chi'(G) = \Delta(G)+1$. The core of $G$, denoted by $G_{\Delta}$, is the subgraph of $G$ induced by all its maximum degree vertices. Hilton and Zhao conjectured that for any critical class 2 graph $G$ with $\Delta(G) \ge 4$, if the maximum degree of $G_{\Delta}$ is at most two, then $G$ is overfull, which in turn gives $\Delta(G) > n/2 +1$. We show that for any critical class 2 graph $G$, if the minimum degree of $G_{\Delta}$ is at most two and $\Delta(G) > n/2 +1$, then $G$ is overfull.