Discussiones Mathematicae Graph Theory 31(2) (2011)
|Csilla Bujtás1 Zsolt Tuza1,2 Vitaly Voloshin3
1Department of Computer Science and Systems Technology
2Computer and Automation Institute
3Department of Mathematics, Physics,
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.
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Received 23 November 2009
Revised 14 July 2010
Accepted 14 July 2010