SIAM Journal on Discrete Mathematics, Volume 35, Issue 2, Page 1478-1502, January 2021.
Let $P_G(k)$ be the number of proper $k$-colorings of a finite simple graph $G$. Tomescu's conjecture, which was recently solved by Fox, He, and Manners, states that $P_G(k) \le k!(k-1)^{n-k}$ for all connected graphs $G$ on $n$ vertices with chromatic number $k\geq 4$. In this paper, we study the same problem with the additional constraint that $G$ is $\ell$-connected. For $2$-connected graphs $G$, we prove a tight bound $P_G(k) \le (k-1)!((k-1)^{n-k+1} + (-1)^{n-k})$ and show that equality is only achieved if $G$ is a $k$-clique with an ear attached. For $\ell \ge 3$, we prove an asymptotically tight upper bound $ P_G(k) \le k!(k-1)^{n-\ell - k + 1} + O((k-2)^n)$ and provide a matching lower bound construction. For the ranges $k \geq \ell$ or $\ell \geq (k-2)(k-1)+1$ we further find the unique graph maximizing $P_G(k)$. We also consider generalizing $\ell$-connected graphs to connected graphs with minimum degree $\delta$.

中文翻译：

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Tomescu's Graph Coloring Conjecture for $\ell$-Connected Graphs

SIAM 离散数学杂志，第 35 卷，第 2 期，第 1478-1502 页，2021 年 1 月。
令 $P_G(k)$ 是有限简单图 $G$ 的正确 $k$-colorings 的数量。最近由 Fox、He 和 Manners 解决的 Tomescu 猜想指出 $P_G(k) \le k!(k-1)^{nk}$ 对于 $n$ 顶点上的所有连通图 $G$数字 $k\geq 4$。在本文中，我们使用 $G$ 是 $\ell$-connected 的附加约束来研究相同的问题。对于$2$-连通图$G$，我们证明了紧界$P_G(k) \le (k-1)!((k-1)^{n-k+1} + (-1)^{nk })$ 并表明只有当 $G$ 是一个带有耳朵的 $k$-clique 时才能实现平等。对于 $\ell \ge 3$，我们证明了一个渐近紧的上界 $ P_G(k) \le k!(k-1)^{n-\ell - k + 1} + O((k-2)^ n)$ 并提供匹配的下界构造。对于 $k \geq \ell$ 或 $\ell \geq (k-2)(k-1)+1$ 的范围，我们进一步找到了最大化 $P_G(k)$ 的唯一图。