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Approximately Coloring Graphs Without Long Induced Paths

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Graph-Theoretic Concepts in Computer Science (WG 2017)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 10520))

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Abstract

It is an open problem whether the 3-coloring problem can be solved in polynomial time in the class of graphs that do not contain an induced path on t vertices, for fixed t. We propose an algorithm that, given a 3-colorable graph without an induced path on t vertices, computes a coloring with \(\max \left\{ 5,2\left\lceil \frac{t-1}{2}\right\rceil -2\right\} \) many colors. If the input graph is triangle-free, we only need \(\max \left\{ 4,\left\lceil \frac{t-1}{2}\right\rceil +1\right\} \) many colors. The running time of our algorithm is \(O((3^{t-2}+t^2)m+n)\) if the input graph has n vertices and m edges.

M. Chudnovsky—Supported by NSF grant DMS-1550991.

S. Spirkl—Supported by Fondecyt grant 1140766 and Millennium Nucleus Information and Coordination in Networks.

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Acknowledgments

We are thankful to Paul Seymour for many helpful discussions. We thank Stefan Hougardy for pointing out [15] to us.

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Authors and Affiliations

  1. Princeton University, Princeton, NJ, 08544, USA

    Maria Chudnovsky & Sophie Spirkl

  2. RWTH Aachen University, Aachen, Germany

    Oliver Schaudt

  3. Universidad de Chile, Santiago, Chile

    Maya Stein

  4. Columbia University, New York, NY, 10027, USA

    Mingxian Zhong

Authors
  1. Maria Chudnovsky
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  2. Oliver Schaudt
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  3. Sophie Spirkl
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  4. Maya Stein
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  5. Mingxian Zhong
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Corresponding author

Correspondence to Oliver Schaudt .

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Editors and Affiliations

  1. Utrecht University, Utrecht, The Netherlands

    Hans L. Bodlaender

  2. Rheinisch-Westfälische Technische Hochschule Aachen, Aachen, Germany

    Gerhard J. Woeginger

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Chudnovsky, M., Schaudt, O., Spirkl, S., Stein, M., Zhong, M. (2017). Approximately Coloring Graphs Without Long Induced Paths. In: Bodlaender, H., Woeginger, G. (eds) Graph-Theoretic Concepts in Computer Science. WG 2017. Lecture Notes in Computer Science(), vol 10520. Springer, Cham. https://doi.org/10.1007/978-3-319-68705-6_15

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  • DOI: https://doi.org/10.1007/978-3-319-68705-6_15

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-68704-9

  • Online ISBN: 978-3-319-68705-6

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