Colouring Planar Mixed Hypergraphs

Abstract

A mixed hypergraph is a triple ${\cal H} = (V,{\cal C}, {\cal D})\;$ where $V$ is the vertex set and ${\cal C}$ and ${\cal D}$ are families of subsets of $V$, the ${\cal C}$-edges and ${\cal D}$-edges, respectively. A $k$-colouring of ${\cal H}$ is a mapping $c: V\rightarrow [k]$ such that each ${\cal C}$-edge has at least two vertices with a ${\cal C}$ommon colour and each ${\cal D}$-edge has at least two vertices of ${\cal D}$ifferent colours. ${\cal H}$ is called a planar mixed hypergraph if its bipartite representation is a planar graph. Classic graphs are the special case of mixed hypergraphs when ${\cal C}=\emptyset$ and all the ${\cal D}$-edges have size 2, whereas in a bi-hypergraph ${\cal C} = {\cal D}$. 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.