Gallai colorings and domination in multipartite digraphs

  • Authors:

  • András Gyárfás;Gábor Simonyi;Ágnes Tóth

  • Affiliations:

  • Computer and Automation Research Institute, Hungarian Academy of Sciences, 1518 BudapestP. O. Box 63, Hungary;Alfréd Rényi Institute of Mathematics, Hungarian Academy of Sciences, 1364 BudapestP. O. BOX 127, Hungary;Department of Computer Science and Information Theory, Budapest University of Technology and Economics, 1521 BudapestP. O. Box 91, Hungary

  • Venue:
  • Journal of Graph Theory
  • Year:
  • 2012

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Abstract

Assume that D is a digraph without cyclic triangles and its vertices are partitioned into classes A1, …, At of independent vertices. A set is called a dominating set of size |S| if for any vertex there is a w∈U such that (w, v)∈E(D). Let β(D) be the cardinality of the largest independent set of D whose vertices are from different partite classes of D. Our main result says that there exists a h = h(β(D)) such that D has a dominating set of size at most h. This result is applied to settle a problem related to generalized Gallai colorings, edge colorings of graphs without 3-colored triangles. © 2011 Wiley Periodicals, Inc. J Graph Theory © 2012 Wiley Periodicals, Inc.