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Coloring Graphs Characterized by a Forbidden Subgraph

  • Petr A. Golovach
  • Daniël Paulusma
  • Bernard Ries
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7464)

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.

Keywords

Polynomial Time Maximum Degree Adjacent Vertex Coloring Problem Graph Class 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Petr A. Golovach
    • 1
  • Daniël Paulusma
    • 2
  • Bernard Ries
    • 3
  1. 1.Department of InformaticsUniversity of BergenNorway
  2. 2.School of Engineering and Computing SciencesDurham UniversityUK
  3. 3.Laboratoire d’Analyse et Modélisation de Systèmes pour l’Aide à la DecisionUniversité Paris DauphineFrance

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