A new improvement to the Overfull Conjecture

Abstract

Let $G$ be a simple graph with order $n$, maximum degree $\D(G)$, minimum degree $\hat{I}´(G)$ and chromatic index $\ddot{I}{‡}^{\prime }(G)$, respectively. A graph $G$ is called {\em $\D$-critical} if $χ'(G)=\D(G)+1$ and $χ'(H)\textless χ'(G)$ for every proper subgraph $H$ of $G$, and $G$ is overfull if $\left|E(G)\right|>\hat{I}”(G)\lfloor n/2\rfloor$. In 1986, Chetwynd and Hilton proposed the Overfull Conjecture: Every $\D$-critical graph $G$ with $\D(G)\textgreater\frac{n}{3}$ is overfull. The Overfull Conjecture has many implications, such as that it implies a polynomial-time algorithm for determining the chromatic index of graphs $G$ with $\D(G)\textgreater\frac{n}{3}$, and implies several longstanding conjectures in the area of graph edge coloring. Recently, Cao, Chen, Jing and Shan (SIAM J. Discrete Math. 2022) verified the Overfull Conjecture for $\D(G)-7δ(G)/4\ge (3n-17)/4$. In this paper, we improve it for $\D(G)-5δ(G)/3\ge (2n-7)/3$.