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.