Overfullness of edge-critical graphs with small minimal core degree

TL;DR

This paper investigates conditions under which edge-critical graphs with small minimal core degree are overfull, contributing to the understanding of the overfull conjecture in graph theory.

Contribution

It introduces new degree conditions involving the core of the graph that guarantee overfullness in critical graphs, advancing the approach to the overfull conjecture.

Findings

Graphs with certain degree bounds are overfull

Critical graphs with small core degree are overfull under specified conditions

Provides new criteria related to the core for overfullness

Abstract

Let $G$ be a simple graph. Denote by $n$, $\Delta(G)$ and $\chi' (G)$ be the order, the maximum degree and the chromatic index of $G$, respectively. We call $G$ \emph{overfull} if $|E(G)|/\lfloor n/2\rfloor > \Delta(G)$, and {\it 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 \emph{core} of $G$, denoted by $G_{\Delta}$, is the subgraph of $G$ induced by all its maximum degree vertices. We believe that utilizing the core degree condition could be considered as an approach to attacking the overfull conjecture. Along this direction, we in this paper show that for any integer $k\geq 2$, if $G$ is critical with $\Delta(G)\geq \frac{2}{3}n+\frac{3k}{2}$ and $\delta(G_\Delta)\leq k$, then $G$ is overfull.