The upper chromatic number (Formula presented.) of a hypergraph (Formula presented.) is the maximum number of colors that can occur in a vertex coloring φ:X→N such that no edge E∈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 χ¯, unless P=NP. In sharp contrast to this, here we prove that if the input is restricted to hypertrees H of bounded maximum vertex degree, then χ¯(H) can be determined in linear time if an underlying tree is also given in the input. Consequently, χ¯ on hypertrees is fixed parameter tractable in terms of maximum degree.
- Arboreal hypergraph
- Upper chromatic number
- Vertex coloring
ASJC Scopus subject areas
- Management Science and Operations Research