A graph $G$ is \textit{$k$-critical} if $\chi(G) = k$ and every proper subgraph of $G$ is $(k - 1)$-colorable, and if $L$ is a list-assignment for $G$, then $G$ is \textit{$L$-critical} if $G$ is not $L$-colorable but every proper induced subgraph of $G$ is. In 2014, Kostochka and Yancey proved a lower bound on the average degree of an $n$-vertex $k$-critical graph tending to $k - \frac{2}{k - 1}$ for large $n$ that is tight for infinitely many values of $n$, and they asked how their bound may be improved for graphs not containing a large clique. Answering this question, we prove that for $\varepsilon \leq 2.6\cdot10^{-10}$, if $k$ is sufficiently large and $G$ is a $K_{\omega + 1}$-free $L$-critical graph where $\omega \leq k - \log^{10}k$ and $L$ is a list-assignment for $G$ such that $|L(v)| = k - 1$ for all $v\in V(G)$, then the average degree of $G$ is at least $(1 + \varepsilon)(k - 1) - \varepsilon \omega - 1$. This result implies that for some $\varepsilon > 0$, for every graph $G$ satisfying $\omega(G) \leq \mathrm{mad}(G) - \log^{10}\mathrm{mad}(G)$ where $\omega(G)$ is the size of the largest clique in $G$ and $\mathrm{mad}(G)$ is the maximum average degree of $G$, the list-chromatic number of $G$ is at most $\left\lceil (1 - \varepsilon)(\mathrm{mad}(G) + 1) + \varepsilon\omega(G)\right\rceil$.

翻译：图$G$称为\textit{$k$-临界图}若$\chi(G)=k$且$G$的每个真子图均为$(k-1)$-可着色；若$L$是$G$的一个列表分配，则$G$称为\textit{$L$-临界图}若$G$不是$L$-可着色的但每个真导出子图均为$L$-可着色。2014年，Kostochka与Yancey证明了$n$顶点$k$-临界图的平均度下界趋向于$k - \frac{2}{k-1}$（当$n$充分大时），该界对无穷多个$n$是紧的，并提出了一个开放问题：如何在不含大团的图中改进这一下界。回答该问题，我们证明：对于$\varepsilon \leq 2.6\times10^{-10}$，若$k$充分大，$G$是无$K_{\omega+1}$的$L$-临界图（其中$\omega \leq k - \log^{10}k$），且$L$是$G$的列表分配满足对所有$v\in V(G)$有$|L(v)| = k-1$，则$G$的平均度至少为$(1+\varepsilon)(k-1) - \varepsilon\omega - 1$。这一结果蕴含：存在某个$\varepsilon>0$，使得对每个满足$\omega(G) \leq \mathrm{mad}(G) - \log^{10}\mathrm{mad}(G)$的图$G$（其中$\omega(G)$是$G$的最大团大小，$\mathrm{mad}(G)$是$G$的最大平均度），其列表色数至多为$\left\lceil (1 - \varepsilon)(\mathrm{mad}(G) + 1) + \varepsilon\omega(G)\right\rceil$。