Let G be a simple graph, and let ${\chi }^{\prime }(G)$ and $\mathrm{\Delta }(G)$ denote the chromatic index and the maximum degree of G, respectively. A graph G is a critical class two graph if ${\chi }^{\prime }(G)=\mathrm{\Delta }(G)+1$ and ${\chi }^{\prime }(H)\le \mathrm{\Delta }(G)$ for every proper subgraph H of G. Let $\bar{d}(G)$ denote the average degree of G, i.e., $\bar{d}(G)=2|E(G)|/|V(G)|$. Vizing in 1968 conjectured that $\bar{d}(G)\ge \mathrm{\Delta }(G)-1+3/n$ if G is a critical class two graph of order n. In this paper, we prove that $\bar{d}(G)\ge \frac{3}{4}\mathrm{\Delta }(G)-8$ if G is a critical class two graph. Let $\delta (G)$ denote the minimum degree of G. We show that there exist two functions D and $D_0$ such that for any $ϵ\in (0,1)$, if G is a critical class two graph with $\mathrm{\Delta }(G)\ge D(ϵ)$ and $\delta (G)\ge D_0(ϵ)$, then $\bar{d}(G)\ge (1-ϵ)\mathrm{\Delta }(G)$. We will give two specific functions satisfying the statement above in the paper. Moreover, we show that if G is a critical class two graph and $\delta (G)\ge {(\log \mathrm{\Delta }(G))}^{\frac{3}{4}}$, then $\bar{d}(G)\ge \mathrm{\Delta }(G)-o(\mathrm{\Delta }(G))$.

令G为简单图形，令${\chi }^{\prime }（G）$ 和 $\mathrm{\Delta }（G）$表示色指数和最大程度的ģ分别。的曲线图G ^是一个关键的类中的两个曲线图，如果${\chi }^{\prime }（G）=\mathrm{\Delta }（G）+\mathrm{1个}$ 和 ${\chi }^{\prime }（H）\le \mathrm{\Delta }（G）$为每个子图正确ħ的ģ。让$\stackrel{〜}{d}（G）$表示平均度的G ^，即$\stackrel{〜}{d}（G）=2|Ë（G）|/|V（G）|$。1968年的Vizing推测$\stackrel{〜}{d}（G）\ge \mathrm{\Delta }（G）-\mathrm{1个}+3/ñ$如果G是n阶的临界二类图。在本文中，我们证明$\stackrel{〜}{d}（G）\ge \frac{3}{4}\mathrm{\Delta }（G）-8$如果G是关键的二级图。让$\delta （G）$表示最小程度的ģ。我们证明存在两个函数D和$d_0$ 这样对于任何 $ϵ\in （0，\mathrm{1个}）$，如果G是具有$\mathrm{\Delta }（G）\ge d（ϵ）$ 和 $\delta （G）\ge d_0（ϵ）$， 然后 $\stackrel{〜}{d}（G）\ge （\mathrm{1个}-ϵ）\mathrm{\Delta }（G）$。我们将提供两个特定功能来满足本文中的上述要求。而且，我们证明了如果G是一个关键类2图$\delta （G）\ge {（\mathrm{日志}\mathrm{\Delta }（G））}^{\frac{3}{4}}$， 然后 $\stackrel{〜}{d}（G）\ge \mathrm{\Delta }（G）-Ø（\mathrm{\Delta }（G））$。