List Coloring in the Absence of a Linear Forest

  • Jean-François Couturier
  • Petr A. Golovach
  • Dieter Kratsch
  • Daniël Paulusma
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6986)

Abstract

The k-Coloring problem is to decide whether a graph can be colored with at most k colors such that no two adjacent vertices receive the same color. The List k-Coloring problem requires in addition that every vertex u must receive a color from some given set L(u) ⊆ {1,…,k}. Let Pn denote the path on n vertices, and G + H and rH the disjoint union of two graphs G and H and r copies of H, respectively. For any two fixed integers k and r, we show that List k-Coloring can be solved in polynomial time for graphs with no induced rP1 + P5, hereby extending the result of Hoàng, Kamiński, Lozin, Sawada and Shu for graphs with no induced P5. Our result is tight; we prove that for any graph H that is a supergraph of P1 + P5 with at least 5 edges, already List 5-Coloring is NP-complete for graphs with no induced H. We also show that List k-Coloring is fixed parameter tractable in k + r on graphs with no induced rP1 + P2, and that k-Coloring restricted to such graphs allows a polynomial kernel when parameterized by k. Finally, we show that List k-Coloring is fixed parameter tractable in k for graphs with no induced P1 + P3.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Jean-François Couturier
    • 1
  • Petr A. Golovach
    • 2
  • Dieter Kratsch
    • 1
  • Daniël Paulusma
    • 2
  1. 1.Laboratoire d’Informatique Théorique et AppliquéeUniversité Paul Verlaine - MetzMetz Cedex 01France
  2. 2.School of Engineering and Computing SciencesDurham University, Science LaboratoriesDurhamUnited Kingdom

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