Packing list‐colorings

Abstract

AbstractList coloring is an influential and classic topic in graph theory. We initiate the study of a natural strengthening of this problem, where instead of one list‐coloring, we seek many in parallel. Our explorations have uncovered a potentially rich seam of interesting problems spanning chromatic graph theory. Given a ‐list‐assignment of a graph , which is the assignment of a list of colors to each vertex , we study the existence of pairwise‐disjoint proper colorings of using colors from these lists. We may refer to this as a list‐packing. Using a mix of combinatorial and probabilistic methods, we set out some basic upper bounds on the smallest for which such a list‐packing is always guaranteed, in terms of the number of vertices, the degeneracy, the maximum degree, or the (list) chromatic number of . (The reader might already find it interesting that such a minimal is well defined.) We also pursue a more focused study of the case when is a bipartite graph. Our results do not yet rule out the tantalising prospect that the minimal above is not too much larger than the list chromatic number. Our study has taken inspiration from study of the strong chromatic number, and we also explore generalizations of the problem above in the same spirit.