  A simple graph $G$ is \emph{overfull} if
$|E(G)|>\Delta\lfloor|V(G)|/2\rfloor$. By the pigeonhole principle, every
overfull graph $G$ has $\chi'(G)>\Delta$. The \emph{core} of a graph, denoted
$G_\Delta$, is the subgraph induced by its vertices of degree $\Delta$.
Vizing's Adjacency Lemma implies that if $\chi'(G)>\Delta$, then $G_\Delta$
contains cycles. Hilton and Zhao conjectured that if $G_\Delta$ has maximum
degree 2 and $\Delta\ge 4$, then $\chi'(G)>\Delta$ precisely when $G$ is
overfull. We prove this conjecture for the case $\Delta=4$.
