  Let $G$ be a simple graph. Denote by $n$, $\Delta(G)$ and $\chi' (G)$ be the
order, the maximum degree and the chromatic index of $G$, respectively. We call
$G$
  \emph{overfull} if $|E(G)|/\lfloor n/2\rfloor > \Delta(G)$, and {\it
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 \emph{core} of
$G$, denoted by $G_{\Delta}$, is the subgraph of $G$ induced by all its maximum
degree vertices. We believe that utilizing the core degree condition could be
considered as an approach to attacking the overfull conjecture. Along this
direction, we in this paper show that for any integer $k\geq 2$, if $G$ is
critical with $\Delta(G)\geq \frac{2}{3}n+\frac{3k}{2}$ and
$\delta(G_\Delta)\leq k$, then $G$ is overfull.
