In the study of graph edge coloring for simple graphs, a graph G is called Δ-critical if $\mathrm{\Delta }(G)=\mathrm{\Delta }$, ${\chi }^{\prime }(G)=\mathrm{\Delta }(G)+1$ and ${\chi }^{\prime }(H)<{\chi }^{\prime }(G)$ for every proper subgraph H of G. In this paper, we prove a new adjacency result of critical graphs which allows us to control the degree of vertices with distance four. Combining this result with a previous theorem proved by the authors, we show that for every $ϵ>0$, if G is a Δ-critical graph with order n, then the average degree $\bar{d}(G)\ge (1-ϵ)\mathrm{\Delta }$ and the independence number $\alpha (G)\le (\frac{1}{2}+ϵ)n$ provided Δ is sufficiently large. This shows that, for a Δ-critical graph G, $\bar{d}(G)\ge \mathrm{\Delta }-o(\mathrm{\Delta })$ and $\alpha (G)\le (1/2+o(1))n$ as $\mathrm{\Delta }\to \infty$.

在对简单图的图边缘着色的研究中，图G被称为Δ临界$\mathrm{\Delta }（G）=\mathrm{\Delta }$， ${\chi }^{\prime }（G）=\mathrm{\Delta }（G）+\mathrm{1个}$ 和 ${\chi }^{\prime }（H）<{\chi }^{\prime }（G）$为每个子图正确ħ的ģ。在本文中，我们证明了临界图的新邻接结果，该结果使我们能够控制距离为4的顶点的度。将结果与作者证明的先前定理相结合，我们证明了对于每个$ϵ>0$，如果G是一个n阶的Δ临界图，则平均度$\stackrel{〜}{d}（G）\ge （\mathrm{1个}-ϵ）\mathrm{\Delta }$ 和独立编号 $\alpha （G）\le （\frac{\mathrm{1个}}{2}+ϵ）ñ$只要Δ足够大。这表明，对于Δ临界图G，$\stackrel{〜}{d}（G）\ge \mathrm{\Delta }-Ø（\mathrm{\Delta }）$ 和 $\alpha （G）\le （\mathrm{1个}/2+Ø（\mathrm{1个}））ñ$ 如 $\mathrm{\Delta }\to \infty$。