The overfullness of graphs with small minimum degree and large maximum degree

TL;DR

This paper investigates conditions under which graphs with small minimum degree and large maximum degree are overfull, contributing to the overfull conjecture by establishing a new degree-based criterion.

Contribution

The paper provides a new sufficient condition involving maximum and minimum degrees for a graph to be overfull, advancing understanding of the overfull conjecture.

Findings

Proves that $ riangle$-critical graphs are overfull under certain degree conditions.

Establishes a new inequality involving maximum and minimum degrees that guarantees overfullness.

Supports the overfull conjecture for graphs with specific degree constraints.

Abstract

Given a simple graph $G$, denote by $\Delta(G)$, $\delta(G)$, and $\chi'(G)$ the maximum degree, the minimum degree, and the chromatic index of $G$, respectively. We say $G$ is \emph{$\Delta$-critical} if $\chi'(G)=\Delta(G)+1$ and $\chi'(H)\le \Delta(G)$ for every proper subgraph $H$ of $G$; and $G$ is \emph{overfull} if $|E(G)|>\Delta \lfloor |V(G)|/2 \rfloor$. Since a maximum matching in $G$ can have size at most $\lfloor |V(G)|/2 \rfloor$, it follows that $\chi'(G) = \Delta(G) +1$ if $G$ is overfull. Conversely, let $G$ be a $\Delta$-critical graph. The well known overfull conjecture of Chetwynd and Hilton asserts that $G$ is overfull provided $\Delta(G) > |V(G)|/3$. In this paper, we show that any $\Delta$-critical graph $G$ is overfull if $\Delta(G) - 7\delta(G)/4\ge(3|V(G)|-17)/4$.