Overfullness of critical class 2 graphs with a small core degree

Abstract: Let $G$ be a simple graph, and let $n$, $\Delta(G)$ and $\chi' (G)$ be the order, the maximum degree and the chromatic index of $G$, respectively. We call $G$ overfull if $|E(G)|/\lfloor n/2\rfloor > \Delta(G)$, and critical if $\chi'(H) < \chi'(G)$ for every proper subgraph $H$ of $G$. Clearly, if $G$ is overfull then $\chi'(G) = \Delta(G)+1$. The core of $G$, denoted by $G_{\Delta}$, is the subgraph of $G$ induced by all its maximum degree vertices. Hilton and Zhao conjectured that for any critical class 2 graph $G$ with $\Delta(G) \ge 4$, if the maximum degree of $G_{\Delta}$ is at most two, then $G$ is overfull, which in turn gives $\Delta(G) > n/2 +1$. We show that for any critical class 2 graph $G$, if the minimum degree of $G_{\Delta}$ is at most two and $\Delta(G) > n/2 +1$, then $G$ is overfull.