A new improvement to the Overfull Conjecture

Let $G$ be a simple graph with order $n$, maximum degree $\text{D}(G)$, minimum degree $\delta (G)$ and chromatic index ${\chi }^{\mathrm{\prime }}(G)$, respectively. A graph $G$ is called {\em $\text{D}$-critical} if ${\chi }^{\mathrm{\prime }}(G)=\text{D}(G)+1$ and ${\chi }^{\mathrm{\prime }}(H)\text{textless}{\chi }^{\mathrm{\prime }}(G)$ for every proper subgraph $H$ of $G$, and $G$ is overfull if $\mid E(G)\mid >\mathrm{\Delta }(G)\lfloor n\mathrm{/}2\rfloor$. In 1986, Chetwynd and Hilton proposed the Overfull Conjecture: Every $\text{D}$-critical graph $G$ with $\text{D}(G)\text{textgreater}\frac{n}{3}$ 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 $\text{D}(G)\text{textgreater}\frac{n}{3}$, 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 $\text{D}(G)-7\delta (G)\mathrm{/}4\ge (3n-17)\mathrm{/}4$. In this paper, we improve it for $\text{D}(G)-5\delta (G)\mathrm{/}3\ge (2n-7)\mathrm{/}3$.