On Complexity of Colouring Mixed Hypertrees

Abstract

A mixed hypergraph is a triple $\left( {V,\mathcal{C},\mathcal{D}} \right)$ where V is its vertex set and $\mathcal{C}$ and $\mathcal{D}$ are families of subsets of V, $\mathcal{C}$-edges and $\mathcal{D}$-edges. We demand in a proper colouring that each $\mathcal{C}$-edge contains two vertices with the same colour and each $\mathcal{D}$-edge contains two vertices with different colours. A hypergraph is a hypertree if there exists a tree such that the edges of the hypergraph induce connected subgraphs of that tree.

We prove that it is NP-complete to decide existence of a proper k-colouring even for mixed hypertrees with $\mathcal{C} = \mathcal{D}$ when k is given as part of input. We present a polynomial-time algorithm for colouring mixed hypertrees on trees of bounded degree with fixed number of colours.

Supported in part by GAČR 201/1999/0242, GAUK 158/1999 and KONTAKT 338/99.

Supported by Ministry of Education of Czech Republic as project LN00A056