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Canonization of a Random Circulant Graph by Counting Walks

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WALCOM: Algorithms and Computation (WALCOM 2024)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 14549))

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Abstract

It is well known that almost all graphs are canonizable by a simple combinatorial routine known as color refinement. With high probability, this method assigns a unique label to each vertex of a random input graph and, hence, it is applicable only to asymmetric graphs. The strength of combinatorial refinement techniques becomes a subtle issue if the input graphs are highly symmetric. We prove that the combination of color refinement with vertex individualization produces a canonical labeling for almost all circulant digraphs (Cayley digraphs of a cyclic group). To our best knowledge, this is the first application of combinatorial refinement in the realm of vertex-transitive graphs. Remarkably, we do not even need the full power of the color refinement algorithm. We show that the canonical label of a vertex v can be obtained just by counting walks of each length from v to an individualized vertex.

O. Verbitsky was supported by DFG grant KO 1053/8–2. He is on leave from the IAPMM, Lviv, Ukraine.

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Notes

  1. 1.

    Recall that \(\mathbb {Q}(\zeta _n)\) is obtained by adjoining \(\zeta _n\) to the field of rationals \(\mathbb {Q}\). In other words, this is the smallest subfield of \(\mathbb {C}\) containing \(\mathbb {Q}\) and \(\zeta _n\).

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Authors and Affiliations

  1. Institut für Informatik, Humboldt-Universität zu Berlin, Berlin, Germany

    Oleg Verbitsky

  2. Department of Computer Science, University of Sheffield, Sheffield, UK

    Maksim Zhukovskii

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  1. Oleg Verbitsky
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  2. Maksim Zhukovskii
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Correspondence to Oleg Verbitsky .

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Editors and Affiliations

  1. Japan Advanced Institute of Science and Technology, Nomi, Japan

    Ryuhei Uehara

  2. Iwate University, Morioka, Japan

    Katsuhisa Yamanaka

  3. National Taiwan University, Taipei, Taiwan

    Hsu-Chun Yen

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Verbitsky, O., Zhukovskii, M. (2024). Canonization of a Random Circulant Graph by Counting Walks. In: Uehara, R., Yamanaka, K., Yen, HC. (eds) WALCOM: Algorithms and Computation. WALCOM 2024. Lecture Notes in Computer Science, vol 14549. Springer, Singapore. https://doi.org/10.1007/978-981-97-0566-5_23

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  • DOI: https://doi.org/10.1007/978-981-97-0566-5_23

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  • Online ISBN: 978-981-97-0566-5

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