## COLOR-BOUNDED HYPERGRAPHS, V: HOST GRAPHS AND SUBDIVISIONS

 Csilla Bujtás1  Zsolt Tuza1,2  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,...,Em}, together with integers si and ti satisfying 1 ≤ si ≤ ti ≤ |Ei| for each i = 1,...,m. A vertex coloring φ is proper if for every i, the number of colors occurring in edge Ei satisfies si ≤ |φ(Ei)| ≤ ti. 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 Ei 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 |Ei| = 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 |Ei| ≤ 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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