Discussiones Mathematicae Graph Theory 31(2) (2011)
223-238
doi: 10.7151/dmgt.1541
Csilla Bujtás^{1} Zsolt Tuza^{1,2} Vitaly Voloshin^{3}
^{1}Department of Computer Science and Systems Technology ^{2}Computer and Automation Institute ^{3}Department 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 E_{i} induces a connected subgraph in G. In the current setting we fix a graph or multigraph G_{0}, and assume that the host graph G is obtained by some sequence of edge subdivisions, starting from G_{0}.
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 G_{0} 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 G_{0} by edge subdivisions. Stronger bounds are derived for hypergraphs for which G_{0} 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