AI 中文总结
本文研究列表临界图与对应临界图的密度，证明了相关函数的下界，改进了布鲁克斯定理，结果在对应着色语境下成立，还得到了反函数约束满足问题的相关推论。
AI 中文摘要
若图G不是(k-1)-列表可着色的，但其每个真子图都是(k-1)-列表可着色的，则称G是列表k-临界图。本文研究了两个函数：一是表示n个顶点的列表k-临界图的最小边数的函数fₗ(n,k)，二是gₗ(k) = liminfₙ→∞ [2/n (fₗ(n,k) - k + 1)]。我们证明，对于所有k ≥ 4且n ≥ k+2，每个有n ≥ k+2个顶点的列表k-临界图的边数都超过(k-1 + 1/28)·n/2，这意味着对于所有k ≥ 4，gₗ(k) ≥ 1/28。这是首个表明当k→∞时liminf gₗ(k) > 0的结果。我们还证明，对于所有k ≥ 352，gₗ(k) ≥ 1/24。作为我们结果的推论，我们得到了布鲁克斯定理的如下改进：对于所有d ≥ 3，若图G不含K_{d+1}子图且最大平均度不超过d + 1/28，则G是d-列表可着色的。我们的所有结果在对应着色（即DP-着色）的语境下同样成立。作为对应着色结果的一个推论，我们还证明，对于每个d ≥ 3，变量域大小为d的极小不可满足反函数约束满足问题（CSP），其原始图要么包含K_{d+1}，要么平均度至少为d + 1/28。
英文摘要
A graph $G$ is list $k$-critical if $G$ is not $(k-1)$-list-colorable, but every proper subgraph of $G$ is $(k-1)$-list-colorable. In this paper, we study the function $f_{\ell}(n,k)$ denoting the minimum number of edges in an $n$-vertex list $k$-critical graph, as well as the function $g_{\ell}(k) = \liminf_{n \rightarrow \infty} \frac 2n (f_{\ell}(n,k) - k + 1)$. We show that for all $k \geq 4$ and $n \geq k+2$, every list $k$-critical graph on $n \geq k+2$ vertices has more than $(k-1+\frac 1{28}) \frac n2$ edges, which implies that $g_{\ell}(k) \geq \frac{1}{28}$ for all $k \geq 4$. This is the first result showing that $\liminf_{k \rightarrow \infty} g_{\ell}(k) > 0$. We also show that $g_{\ell}(k) \geq \frac{1}{24}$ for all $k \geq 352$. As a corollary to our result, we obtain the following improvement to Brooks' theorem: For all $d \geq 3$, if $G$ has no $K_{d+1}$ subgraph and has maximum average degree at most $d+\frac 1{28}$, then $G$ is $d$-list-colorable. All of our results hold in the setting of correspondence coloring (DP-coloring) as well. As a corollary of our correspondence coloring result, we also show that for each $d \geq 3$, a minimal unsatisfiable anti-functional constraint satisfaction problem (CSP) with variable domains of size $d$ has a primal graph either containing $K_{d+1}$ or with average degree at least $d+\frac 1{28}$.
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