List Packing and Correspondence Packing of Planar Graphs

For a graph $G$ and a list assignment $L$ with $\mid L\left(v\right)\mid =k$ for all $v$, an $L$-packing consists of $L$-colorings ${\phi }_1,\dots ,{\phi }_k$ such that ${\phi }_i\left(v\right)\ne {\phi }_j\left(v\right)$ for all $v$ and all distinct $i,j\in \left\{1,\dots ,k\right\}$. Let ${\chi }_{\ell }^{\star }\left(G\right)$ denote the smallest $k$ such that $G$ has an $L$-packing for every $L$ with $\mid L\left(v\right)\mid =k$ for all $v$. Let $P_k$ denote the set of all planar graphs with girth at least $k$. We show that (i) ${\chi }_{\ell }^{\star }\left(G\right)⩽8$ for all $G\in P_3$ and (ii) ${\chi }_{\ell }^{\star }\left(G\right)⩽5$ for all $G\in P_4$ and (iii) ${\chi }_{\ell }^{\star }\left(G\right)⩽4$ for all $G\in P_5$. Part (i) makes progress on a problem of Cambie, Cames van Batenburg, Davies, and Kang. We also consider the analogue of ${\chi }_{\ell }^{\star }$ for correspondence coloring, ${\chi }_c^{\star }$. In fact, all bounds stated above for ${\chi }_{\ell }^{\star }$ also hold for ${\chi }_c^{\star }$.