A Short Proof that the List Packing Number of any Graph is Well Defined

Abstract

List packing is a notion that was introduced in 2021 (by Cambie et al.). The list packing number of a graph $G$, denoted $\chi_{\ell}^*(G)$, is the least $k$ such that for any list assignment $L$ that assigns $k$ colors to each vertex of $G$, there is a set of $k$ proper $L$-colorings of $G$, $\{f_1, \ldots, f_k \}$, with the property $f_i(v) \neq f_j(v)$ whenever $1 \leq i < j \leq k$ and $v \in V(G)$. We present a short proof that for any graph $G$, $\chi_{\ell}^*(G) \leq |V(G)|$. Interestingly, our proof makes use of Galvin's celebrated result that the list chromatic number of the line graph of any bipartite multigraph equals its chromatic number.