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

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