Packing list-colorings

Abstract

List 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 (Formula presented.) -list-assignment (Formula presented.) of a graph (Formula presented.), which is the assignment of a list (Formula presented.) of (Formula presented.) colors to each vertex (Formula presented.), we study the existence of (Formula presented.) pairwise-disjoint proper colorings of (Formula presented.) 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 (Formula presented.) 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 (Formula presented.). (The reader might already find it interesting that such a minimal (Formula presented.) is well defined.) We also pursue a more focused study of the case when (Formula presented.) is a bipartite graph. Our results do not yet rule out the tantalising prospect that the minimal (Formula presented.) 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.