覆盖族用于完全二部图的笛卡尔积的DP-染色
Covering Families for DP-Coloring of Cartesian Products with Complete Bipartite Graphs
- Illinois Institute of Technology（伊利诺伊理工学院）
- University of South Alabama（南阿拉巴马大学）
- Lake Forest College（湖森林学院）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文引入覆盖族概念，证明其最小大小与DP-染色中完全二部图笛卡尔积的临界参数相等，从而证明μ(4)=12并改进上下界。
AI中文摘要：
列表染色中一个著名的民间结果展示了图的列表色数与色数之间的差距可以任意大，即：$\chi_{\ell}(K_{l,t}) = 1+l$ 当且仅当 $t \geq l^l$。DP-染色（也称为对应染色）是列表染色的一个被广泛研究的推广，于2015年引入。2018年，Mudrock研究了上述民间结果的DP类比。他证明对于 $l \in\mathbb{N}$，若 $\mu(l)$ 是使得 $\chi_{DP}(K_{l,t})=1+l$ 的最小整数 $t$，则 $\left\lceil l^l/l!\right\rceil \leq \mu(l) \leq 1+l^l(\log(l!)+1)/l!$。最近，Kaul、Mudrock和Sharma研究了该问题的一个更一般版本，通过研究使得 $\chi_{DP}(G \square K_{l,t}) = k + l$ 的最小 $t$，其中 $G$ 满足某些临界条件，且 $G \square K_{l,t}$ 表示 $G$ 与 $K_{l,t}$ 的笛卡尔积。在本文中，我们引入了一个称为覆盖族的概念，为这些DP-染色问题提供了新的视角。特别地，若 $\kappa(l)$ 表示 $[l]^l$ 的覆盖族的最小大小，我们证明 $\mu(l)=\kappa(l)$。我们利用这一等价性证明了 $\mu(4)=12$，并获得了关于 $\mu(l)$ 的新的一般下界。我们还证明了覆盖族最小大小的一般上界，这改进了 $\mu(l)$ 的一般上界，并对涉及完全二部图笛卡尔积的相关DP-染色问题的已知界给出了改进。
英文摘要：
A famous folklore result in list coloring demonstrating that the gap between the list chromatic number and chromatic number of a graph can be arbitrarily large is: $χ_{\ell}(K_{l,t}) = 1+l$ if and only if $t \geq l^l$. DP-coloring (also called correspondence coloring) is a well-studied generalization of list coloring introduced in 2015. In 2018, Mudrock studied the DP analogue of the aforementioned folklore result. He proved that for $l \in\mathbb{N}$, if $μ(l)$ is the smallest integer $t$ such that $χ_{DP}(K_{l,t})=1+l$, then $\left\lceil l^l/l!\right\rceil \leq μ(l) \leq 1+l^l(\log(l!)+1)/l!$. Recently, Kaul, Mudrock, and Sharma studied a more general version of this problem by studying the smallest $t$ for which $χ_{DP}(G \square K_{l,t}) = k + l$, where $G$ satisfies certain criticality conditions and $G \square K_{l,t}$ denotes the Cartesian product of $G$ and $K_{l,t}$. In this paper, we introduce a notion we call covering families that gives a new perspective on these DP-coloring questions. In particular, if $κ(l)$ denotes the minimum size of a covering family of $[l]^l$, we show that $μ(l)=κ(l)$. We use this equivalence to prove $μ(4)=12$ and to obtain new general lower bounds on $μ(l)$. We also prove a general upper bound on the minimum size of covering families which yields an improved general upper bound on $μ(l)$ and gives improvements on known bounds for related DP-coloring questions involving Cartesian products with complete bipartite graphs.
