Maximum number of colors in hypertrees of bounded degree

Abstract

The upper chromatic number $\overline{\chi }(\mathcal{H})$ of a hypergraph $\mathcal{H}=(X,\mathcal{E})$ is the maximum number of colors that can occur in a vertex coloring $\varphi :X\rightarrow \mathbb {N}$ such that no edge $E\in \mathcal{E}$ is completely multicolored. A hypertree (also called arboreal hypergraph) is a hypergraph whose edges induce subtrees on a fixed tree graph. It has been shown that on hypertrees it is algorithmically hard not only to determine exactly but also to approximate the value of $\overline{\chi }$, unless $\mathsf{P}=\mathsf{NP}$. In sharp contrast to this, here we prove that if the input is restricted to hypertrees $\mathcal{H}$ of bounded maximum vertex degree, then $\overline{\chi }(\mathcal{H})$ can be determined in linear time if an underlying tree is also given in the input. Consequently, $\overline{\chi }$ on hypertrees is fixed parameter tractable in terms of maximum degree.