List Packing and Correspondence Packing of Planar Graphs

Abstract

For a graph $G$ and a list assignment $L$ with $|L(v)|=k$ for all $v$, an $L$-packing consists of $L$-colorings $φ_1,\cdots,φ_k$ such that $φ_i(v)\neφ_j(v)$ for all $v$ and all distinct $i,j\in\{1,\ldots,k\}$. Let $χ^{\star}_{\ell}(G)$ denote the smallest $k$ such that $G$ has an $L$-packing for every $L$ with $|L(v)|=k$ for all $v$. Let $\mathcal{P}_k$ denote the set of all planar graphs with girth at least $k$. We show that (i) $χ^{\star}_{\ell}(G)\le 8$ for all $G\in \mathcal{P}_3$ and (ii) $χ^{\star}_{\ell}(G)\le 5$ for all $G\in \mathcal{P}_4$ and (iii) $χ^{\star}_{\ell}(G)\le 4$ for all $G\in \mathcal{P}_5$. Part (i) makes progress on a problem of Cambie, Cames van Batenburg, Davies, and Kang. We also construct outerplanar graphs $G$ such that $χ^{\star}_{\ell}(G)=4$, which matches the known upper bound $χ^{\star}_{\ell}(G)\le 4$ for all outerplanar graphs. Finally, we consider the analogue of $χ^{\star}_{\ell}$ for correspondence coloring, $χ^{\star}_c$. In fact, all bounds stated above for $χ^{\star}_{\ell}$ also hold for $χ^{\star}_c$.