The Overfullness of Graphs with Small Minimum Degree and Large Maximum Degree

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