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 χℓ⁎(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, {f1,…,fk}, with the property fi(v)≠fj(v) whenever 1≤i<j≤k and v∈V(G). We present a short proof that for any graph G, χℓ⁎(G)≤|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.