Mixed hypercacti

Abstract

A mixed hypergraph H is a triple (V, C, D) where V is its vertex set and C and D are families of subsets of V (called C -edges and D -edges). A vertex coloring of H is proper if each C -edge contains two vertices with the same color and each D -edge contains two vertices with different colors. The feasible set of H is the set of all k's such that there exists a proper coloring using exactly k colors. The feasible set is gap-free if it is an interval of integers. A graph is a strong/weak cactus if all its cycles are vertex/edge-disjoint. A hypergraph is spanned by a graph (with the same vertex set) if the edges of the hypergraph induce connected subgraphs. A strong/weak hypercactus is spanned by a strong/weak cactus. We prove that the feasible set of any mixed strong hypercactus is gap-free. We find infinitely many mixed weak hypercacti such that the feasible set of any of them contains a gap. For each connected non-planar graph G≠ K 5, we find a mixed hypergraph spanned by G whose feasible set contains a gap.