  Given a graph $G$, denote by $\Delta$, $\bar{d}$ and $\chi^\prime$ the
maximum degree, the average degree and the chromatic index of $G$,
respectively. A simple graph $G$ is called {\it edge-$\Delta$-critical} if
$\chi^\prime(G)=\Delta+1$ and $\chi^\prime(H)\le\Delta$ for every proper
subgraph $H$ of $G$. Vizing in 1968 conjectured that if $G$ is
edge-$\Delta$-critical, then $\bar{d}\geq \Delta-1+ \frac{3}{n}$. We show that
$$ \begin{displaystyle} \avd \ge \begin{cases}
  0.69241\D-0.15658 \quad\,\: \mbox{ if } \Delta\geq 66,
  0.69392\D-0.20642\quad\;\,\mbox{ if } \Delta=65, \mbox{ and }
  0.68706\D+0.19815\quad\! \quad\mbox{if } 56\leq \Delta\leq64.
  \end{cases}
  \end{displaystyle}
  $$
  This result improves the best known bound $\frac{2}{3}(\Delta +2)$ obtained
by Woodall in 2007 for $\Delta \geq 56$. Additionally, Woodall constructed an
infinite family of graphs showing his result cannot be improved by well-known
Vizing's Adjacency Lemma and other known edge-coloring techniques. To over come
the barrier, we follow the recently developed recoloring technique of Tashkinov
trees to expand Vizing fans technique to a larger class of trees.
