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On the Power of Color Refinement

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Fundamentals of Computation Theory (FCT 2015)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 9210))

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Abstract

Color refinement is a classical technique used to show that two given graphs G and H are non-isomorphic; it is very efficient, although it does not succeed on all graphs. We call a graph G amenable to color refinement if the color-refinement procedure succeeds in distinguishing G from any non-isomorphic graph H. Babai, Erdős, and Selkow (1982) have shown that random graphs are amenable with high probability. We determine the exact range of applicability of color refinement by showing that amenable graphs are recognizable in time \(O((n+m)\log n)\), where n and m denote the number of vertices and the number of edges in the input graph.

This work was supported by the Alexander von Humboldt Foundation in its research group linkage program. The second author and the fourth author were supported by DFG grants KO 1053/7-2 and VE 652/1-2, respectively.

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Notes

  1. 1.

    The last case, in which the graph is the 5-cycle, is missing from the statement of this result in [11, Theorem 2.12]. The proof in [11] tacitly considers only graphs with at least 6 vertices.

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Author information

Authors and Affiliations

  1. The Institute of Mathematical Sciences, Chennai, 600 113, India

    V. Arvind & Gaurav Rattan

  2. Institut für Informatik, Humboldt Universität zu Berlin, Berlin, Germany

    Johannes Köbler & Oleg Verbitsky

  3. On leave from the Institute for Applied Problems of Mechanics and Mathematics, Lviv, Ukraine

    Oleg Verbitsky

Authors
  1. V. Arvind
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  2. Johannes Köbler
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  3. Gaurav Rattan
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  4. Oleg Verbitsky
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Corresponding author

Correspondence to Johannes Köbler .

Editor information

Editors and Affiliations

  1. Université Paris Diderot, Paris, France

    Adrian Kosowski

  2. Université Bordeaux 1, Talence, France

    Igor Walukiewicz

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Arvind, V., Köbler, J., Rattan, G., Verbitsky, O. (2015). On the Power of Color Refinement. In: Kosowski, A., Walukiewicz, I. (eds) Fundamentals of Computation Theory. FCT 2015. Lecture Notes in Computer Science(), vol 9210. Springer, Cham. https://doi.org/10.1007/978-3-319-22177-9_26

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  • DOI: https://doi.org/10.1007/978-3-319-22177-9_26

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-22176-2

  • Online ISBN: 978-3-319-22177-9

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