# Color-bounded hypergraphs, V: Host graphs and subdivisions

Bujtás, Csilla and Tuza, Zsolt and Voloshin, V
(2011)
*Color-bounded hypergraphs, V: Host graphs and subdivisions.*
DISCUSSIONES MATHEMATICAE GRAPH THEORY, 31 (2).
pp. 223-238.
ISSN 1234-3099

## Abstract

A color-bounded hypergraph is a hypergraph (set system) with ver- tex set X and edge set ε = {E1, . . . ,Em}, together with integers si and ti satisfying 1 ≤ si ≤ ti ≤ |E1| for each i = 1, . . . ,m. A vertex coloring φ is proper if for every i, the number of colors occurring in edge 1 satisfies si ≤ |φ(Ei)| ≤ t i. The hypergraph H is colorable if it admits at least one proper coloring. We consider hypergraphs H over a "host graph", that means a graph G on the same vertex set X as H, such that each 1 induces a connected subgraph in G. In the current setting we fix a graph or multigraph G0, and assume that the host graph G is obtained by some sequence of edge subdivisions, starting from G0. The colorability problem is known to be NP-complete in general, and also when restricted to 3-uniform "mixed hypergraphs", i.e., color- bounded hypergraphs in which |1| = 3 and 1 ≤ si ≤ 2 ≤ ti ≤ 3 holds for all i ≤ m. We prove that for every fixed graph G0 and natural number r, colorability is decidable in polynomial time over the class of r-uniform hypergraphs (and more generally of hypergraphs with |1| ≤ r for all 1 ≤ i ≤ m) having a host graph G obtained from G0 by edge subdivisions. Stronger bounds are derived for hypergraphs for which G0 is a tree.

Item Type: | Article |
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Uncontrolled Keywords: | Vertex col-oring, Mixed hypergraph, hypertree, Host graph, Feasible set, Edge subdivision, Color-bounded hypergraph, Arboreal hypergraph |

Subjects: | Q Science > QA Mathematics and Computer Science > QA75 Electronic computers. Computer science / számítástechnika, számítógéptudomány |

Depositing User: | EPrints Admin |

Date Deposited: | 12 Dec 2012 08:40 |

Last Modified: | 05 Feb 2014 12:16 |

URI: | https://eprints.sztaki.hu/id/eprint/6586 |

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