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.
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Notes
Same as ‘arboreal hypergraph’ in part of the literature.
Same as ‘hitting set’ or ‘vertex cover’.
Also called ‘stable set’, but some papers use the two terms differently for hypergraphs.
In fact, the stronger inequality \( |{\mathfrak {F}}_i| \le {D\atopwithdelims ()\lfloor D/2\rfloor }\) also holds, since we have a Sperner family.
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Acknowledgments
Research supported in part by the Hungarian Scientific Research Fund, OTKA Grant T-81493, moreover by the European Union and Hungary, co-financed by the European Social Fund through the project TÁMOP-4.2.2.C-11/1/KONV-2012-0004—National Research Center for Development and Market Introduction of Advanced Information and Communication Technologies.
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Bujtás, C., Tuza, Z. Maximum number of colors in hypertrees of bounded degree. Cent Eur J Oper Res 23, 867–876 (2015). https://doi.org/10.1007/s10100-014-0357-4
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DOI: https://doi.org/10.1007/s10100-014-0357-4