Covering Families for DP-Coloring of Cartesian Products with Complete Bipartite Graphs

Source abstract

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.