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On Coloring Graphs without Induced Forests

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Algorithms and Computation (ISAAC 2010)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 6507))

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Abstract

The ℓ-Coloring problem is the problem to decide whether a graph can be colored with at most ℓ colors. Let P k denote the path on k vertices and G + H and 2H the disjoint union of two graphs G and H or two copies of H, respectively. We solve a known open problem by showing that 3-Coloring is polynomial-time solvable for the class of graphs with no induced 2P 3. This implies that the complexity of 3-Coloring for graphs with no induced graph H is now classified for any fixed graph H on at most 6 vertices. The Vertex Coloring problem is the problem to determine the chromatic number of a graph. We show that Vertex Coloring is polynomial-time solvable for the class of triangle-free graphs with no induced 2P 3 and for the class of triangle-free graphs with no induced P 2 + P 4. This solves two open problems of Dabrowski, Lozin, Raman and Ries and implies that the complexity of Vertex Coloring for triangle-free graphs with no induced graph H is now classified for any fixed graph H on at most 6 vertices. Our proof technique for the case H = 2P 3 is based on a novel structural result on the existence of small dominating sets in 2P 3-free graphs that admit a k-coloring for some fixed k.

This work has been supported by EPSRC (EP/G043434/1).

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Author information

Authors and Affiliations

  1. School of Engineering and Computing Sciences, Science Laboratories, Durham University, South Road, Durham, DH1 3LE, England

    Hajo Broersma, Petr A. Golovach, Daniël Paulusma & Jian Song

Authors
  1. Hajo Broersma
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  2. Petr A. Golovach
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  3. Daniël Paulusma
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  4. Jian Song
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Editor information

Editors and Affiliations

  1. Department of Computer Science, KAIST, Gwahangno 335, Yuseong-gu, 305-701, Daejeon, Korea

    Otfried Cheong  & Kyung-Yong Chwa  & 

  2. School of Computer Science and Engineering, Seoul National University, 151-742, Seoul, Korea

    Kunsoo Park

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Broersma, H., Golovach, P.A., Paulusma, D., Song, J. (2010). On Coloring Graphs without Induced Forests. In: Cheong, O., Chwa, KY., Park, K. (eds) Algorithms and Computation. ISAAC 2010. Lecture Notes in Computer Science, vol 6507. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-17514-5_14

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  • DOI: https://doi.org/10.1007/978-3-642-17514-5_14

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-17513-8

  • Online ISBN: 978-3-642-17514-5

  • eBook Packages: Computer ScienceComputer Science (R0)

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