Color-bounded hypergraphs, II: Interval hypergraphs and hypertrees

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Abstract

A color-bounded hypergraph is a hypergraph (set system) with vertex set X and edge set E = {E1, ..., Em}, together with integers si and ti (1 ≤ si ≤ ti ≤ | Ei |) for i = 1, ..., m. A vertex coloring φ is feasible if the number of colors occurring in edge Ei satisfies si ≤ | φ (Ei) | ≤ ti, for every i ≤ m. In this paper we point out that hypertrees-hypergraphs admitting a representation over a (graph) tree where each hyperedge Ei induces a subtree of the underlying tree-play a central role concerning the set of possible numbers of colors that can occur in feasible colorings. We also consider interval hypergraphs and circular hypergraphs, where the underlying graph is a path or a cycle, respectively. Sufficient conditions are given for a 'gap-free' chromatic spectrum; i.e., when each number of colors is feasible between minimum and maximum. The algorithmic complexity of colorability is studied, too. Compared with the 'mixed hypergraphs'-where 'D-edge' means (si, ti) = (2, | Ei |), while 'C-edge' assumes (si, ti) = (1, | Ei | - 1)-the differences are rather significant. © 2008 Elsevier B.V. All rights reserved.

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Bujtás, C., & Tuza, Z. (2009). Color-bounded hypergraphs, II: Interval hypergraphs and hypertrees. Discrete Mathematics, 309(22), 6391–6401. https://doi.org/10.1016/j.disc.2008.10.023

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