DMGT

Discussiones Mathematicae Graph Theory 31(2) (2011) 223-238
DOI: https://doi.org/10.7151/dmgt.1541

COLOR-BOUNDED HYPERGRAPHS, V: HOST GRAPHS AND SUBDIVISIONS

Csilla Bujtás¹ Zsolt Tuza^1,2 Vitaly Voloshin³

¹Department of Computer Science and Systems Technology
University of Pannonia
H-8200 Veszprém, Egyetem u. 10, Hungary

²Computer and Automation Institute
Hungarian Academy of Sciences
H-1111 Budapest, Kende u. 13-17, Hungary

³Department of Mathematics, Physics,
Computer Science and Geomatics
Troy University, Troy, AL 36082, USA

Abstract

A color-bounded hypergraph is a hypergraph (set system) with vertex set X and edge set E = {E₁,...,E_m}, together with integers s_i and t_i satisfying 1 ≤ s_i ≤ t_i ≤ |E_i| for each i = 1,...,m. A vertex coloring φ is proper if for every i, the number of colors occurring in edge E_i satisfies s_i ≤ |φ(E_i)| ≤ 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 E_i 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 |E_i| = 3 and 1 ≤ s_i ≤ 2 ≤ t_i ≤ 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 |E_i| ≤ 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.

Keywords: mixed hypergraph, color-bounded hypergraph, vertex coloring, arboreal hypergraph, hypertree, feasible set, host graph, edge subdivision.

2010 Mathematics Subject Classification: 05C15, 05C65.

References

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Received 23 November 2009
Revised 14 July 2010
Accepted 14 July 2010