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

TL;DR

The paper provides a concise proof that the list packing number of any graph is at most its number of vertices, utilizing Galvin's theorem on bipartite multigraphs.

Contribution

It introduces a short, elegant proof establishing the finiteness of the list packing number for all graphs, connecting it with existing results in graph coloring theory.

Findings

List packing number is bounded above by the number of vertices.

The proof leverages Galvin's theorem on bipartite multigraphs.

The result confirms the well-definedness of the list packing number.

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.