Coloring graphs characterized by a forbidden subgraph

  • Authors:

  • Petr A. Golovach;Daniël Paulusma;Bernard Ries

  • Affiliations:

  • Department of Informatics, University of Bergen, Norway;School of Engineering and Computing Sciences, Durham University, UK;Laboratoire d'Analyse et Modélisation de Systèmes pour l'Aide à la Decision, Université Paris Dauphine, France

  • Venue:
  • MFCS'12 Proceedings of the 37th international conference on Mathematical Foundations of Computer Science
  • Year:
  • 2012

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Abstract

The Coloring problem is to test whether a given graph can be colored with at most k colors for some given k, such that no two adjacent vertices receive the same color. The complexity of this problem on graphs that do not contain some graph H as an induced subgraph is known for each fixed graph H. A natural variant is to forbid a graph H only as a subgraph. We call such graphs strongly H-free and initiate a complexity classification of Coloring for strongly H-free graphs. We show that Coloring is NP-complete for strongly H-free graphs, even for k=3, when H contains a cycle, has maximum degree at least five, or contains a connected component with two vertices of degree four. We also give three conditions on a forest H of maximum degree at most four and with at most one vertex of degree four in each of its connected components, such that Coloring is NP-complete for strongly H-free graphs even for k=3. Finally, we classify the computational complexity of Coloring on strongly H-free graphs for all fixed graphs H up to seven vertices. In particular, we show that Coloring is polynomial-time solvable when H is a forest that has at most seven vertices and maximum degree at most four.