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Colorings with Few Colors: Counting, Enumeration and Combinatorial Bounds

  • Petr A. Golovach
  • Dieter Kratsch
  • Jean-Francois Couturier
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6410)

Abstract

We provide exact algorithms for enumeration and counting problems on edge colorings and total colorings of graphs, if the number of (available) colors is fixed and small. For edge 3-colorings the following is achieved: there is a branching algorithm to enumerate all edge 3-colorings of a connected cubic graph in time O *(25n/8). This implies that the maximum number of edge 3-colorings in an n-vertex connected cubic graph is O *(25n/8). Finally, the maximum number of edge 3-colorings in an n-vertex connected cubic graph is lower bounded by 12 n/10. Similar results are achieved for total 4-colorings of connected cubic graphs. We also present dynamic programming algorithms to count the number of edge k-colorings and total k-colorings for graphs of bounded pathwidth. These algorithms can be used to obtain fast exact exponential time algorithms for counting edge k-colorings and total k-colorings on graphs, if k is small.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Petr A. Golovach
    • 1
  • Dieter Kratsch
    • 2
  • Jean-Francois Couturier
    • 2
  1. 1.School of Engineering and Computing SciencesDurham UniversityDurhamUnited Kingdom
  2. 2.Laboratoire d’Informatique Théorique et AppliquéeUniversité Paul Verlaine - MetzMetz Cedex 01France

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