Colouring planar mixed hypergraphs

Abstract

A mixed hypergraph is a triple H = (V, C, D) where V is the vertex set and C and D are families of subsets of V, the C-edges and D-edges, respectively. A k-colouring of H is a mapping c : V → [k] such that each C-edge has at least two vertices with a Common colour and each D-edge has at least two vertices of Different colours. H is called a planar mixed hypergraph if its bipartite representation is a planar graph. Classic graphs are the special case of mixed hypergraphs when C = 0 and all the D-edges have size 2, whereas in a bi-hypergraph C = D. We investigate the colouring properties of planar mixed hypergraphs. Specifically, we show that maximal planar bi-hypergraphs are 2-colourable, find formulas for their chromatic polynomial and chromatic spectrum in terms of 2-factors in the dual, prove that their chromatic spectrum is gap-free and provide a sharp estimate on the maximum number of colours in a colouring.