Mixed hypergraphs and other coloring problems

Abstract

A mixed hypergraph is a triple ( V , C , D ) where V is the vertex set and C and D are families of subsets of V called C -edges and D -edges, respectively. A proper coloring of a mixed hypergraph ( V , C , D ) is a coloring of its vertices such that no C -edge is polychromatic and no D -edge is monochromatic. We show that mixed hypergraphs can be used to efficiently model several graph coloring problems including homomorphisms of simple graphs and multigraphs, circular colorings, ( H , C , ⩽ K ) -colorings, ( H , C , K ) -colorings, locally surjective, locally bijective and locally injective homomorphisms, L ( p , q ) -labelings, the channel assignment problem, T-colorings and generalized T-colorings.