  Let $G$ be a simple graph with maximum degree $\Delta(G)$ and chromatic index
$\chi'(G)$. A classic result of Vizing indicates that either $\chi'(G
)=\Delta(G)$ or $\chi'(G )=\Delta(G)+1$. The graph $G$ is called
$\Delta$-critical if $G$ is connected, $\chi'(G )=\Delta(G)+1$ and for any
$e\in E(G)$, $\chi'(G-e)=\Delta(G)$. Let $G$ be an $n$-vertex $\Delta$-critical
graph. Vizing conjectured that $\alpha(G)$, the independence number of $G$, is
at most $\frac{n}{2}$. The current best result on this conjecture, shown by
Woodall, is that $\alpha(G)<\frac{3n}{5}$. We show that for any given
$\varepsilon\in (0,1)$, there exist positive constants $d_0(\varepsilon)$ and
$D_0(\varepsilon)$ such that if $G$ is an $n$-vertex $\Delta$-critical graph
with minimum degree at least $d_0$ and maximum degree at least $D_0$, then
$\alpha(G)<(\frac{{1}}{2}+\varepsilon)n$. In particular, we show that if $G$ is
an $n$-vertex $\Delta$-critical graph with minimum degree at least $d$ and
$\Delta(G)\ge (d+2)^{5d+10}$, then \[ \alpha(G) < \left. \begin{cases}
\frac{7n}{12}, & \text{if $d= 3$; } \frac{4n}{7}, & \text{if $d= 4$; }
\frac{d+2+\sqrt[3]{(d-1)d}}{2d+4+\sqrt[3]{(d-1)d}}n<\frac{4n}{7}, & \text{if
$d\ge 19$. } \end{cases} \right. \]
