Independence number of edge-chromatic critical graphs

Let $G$ be a simple graph with maximum degree $\mathrm{\Delta }(G)$ and chromatic index ${\chi }^{\mathrm{\prime }}(G)$. A classic result of Vizing indicates that either $\chi’(G )=\Delta(G)$or$\chi’(G )=\Delta(G)+1$\mathrm{.}Thegraph$G$iscalled$\Delta$-criticalif$G$isconnected,$\chi’(G )=\Delta(G)+1$andforany$e\in E(G)$,$\chi’(G-e)=\Delta(G)$\mathrm{.}Let$G$bean$n$-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 $\epsilon \in (0,1)$, there exist positive constants $d_0(\epsilon )$ and $D_0(\epsilon )$ such that if $G$ is an $n$-vertex $\mathrm{\Delta }$-critical graph with minimum degree at least $d_0$ and maximum degree at least $D_0$, then $\alpha (G)<(\frac{1}{2}+\epsilon )n$. In particular, we show that if $G$ is an $n$-vertex $\mathrm{\Delta }$-critical graph with minimum degree at least $d$ and $\mathrm{\Delta }(G)\ge (d+2{)}^{5d+10}$, then $$\alpha (G)<\{\begin{array}{llll}\frac{7n}{12},& \text{if }d=3\text{; }\frac{4n}{7},& \text{if }d=4\text{; }\frac{d+2+\sqrt[3]{(d-1)d}}{2d+4+\sqrt[3]{(d-1)d}}n<\frac{4n}{7},& \text{if }d\ge 19\text{. }\end{array}$$