  For a simple graph $G$, denote by $n$, $\Delta(G)$, and $\chi'(G)$ its order,
maximum degree, and chromatic index, respectively. A connected class 2 graph
$G$ is edge-chromatic critical if $\chi'(G-e)<\Delta(G)+1$ for every edge $e$
of $G$. Define $G$ to be overfull if $|E(G)|>\Delta(G) \lfloor n/2 \rfloor$.
Clearly, overfull graphs are class 2 and any graph obtained from a regular
graph of even order by splitting a vertex is overfull. Let $G$ be an $n$-vertex
connected regular class 1 graph with $\Delta(G) >n/3$. Hilton and Zhao in 1997
conjectured that if $G^*$ is obtained from $G$ by splitting one vertex of $G$
into two vertices, then $G^*$ is edge-chromatic critical, and they verified the
conjecture for graphs $G$ with $\Delta(G)\ge \frac{n}{2}(\sqrt{7}-1)\approx
0.82n$. The graph $G^*$ is easily verified to be overfull, and so the hardness
of the conjecture lies in showing that the deletion of every of its edge
decreases the chromatic index. Except in 2002, Song showed that the conjecture
is true for a special class of graphs $G$ with $\Delta(G)\ge \frac{n}{2}$, no
other progress on this conjecture had been made. In this paper, we confirm the
conjecture for graphs $G$ with $\Delta(G) \ge 0.75n$.
