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Expander spanning subgraphs with large girth

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Abstract

We conjecture that in any finite graph with large Cheeger constant we can delete a proportion of edges in such a way that the remaining graph has large girth and retains good expansion properties. We prove this when the expansion is large enough in terms of the maximum degree. The condition on expansion covers, for example, large random d-regular graphs. Our proof relies on the Lovász Local Lemma.

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Acknowledgment

The authors thank Gábor Pete for encouraging this collaboration, to Elad Tzalik for his suggestions to improve the paper and to Merav Parter for her comments on possible applications.

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Authors and Affiliations

  1. Department of Mathematics, The Weizmann Institute od Science, Rehovot, 76100, Israel

    Itai Benjamini

  2. Department of Mathematics, University of Chicago, 5734 S. University Avenue, Chicago, IL, 60637, USA

    Mikolaj Fraczyk

  3. Alfréd Rényi Institute of Mathematics, Reáltanoda u. 13-15, H-1053, Budapest, Hungary

    Gábor Kun

  4. Institute of Mathematics, Eötvös Lóránd University, Pázmány Péter sétány 1/c, H-1117, Budapest, Hungary

    Gábor Kun

Authors
  1. Itai Benjamini
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  2. Mikolaj Fraczyk
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  3. Gábor Kun
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Corresponding author

Correspondence to Itai Benjamini.

Additional information

To Benjy Weiss

The first author thanks the Israeli Science Foundation for support.

The third author’s work on the project leading to this application has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No. 741420), from the ÚNKP-20-5 New National Excellence Program of the Ministry of Innovation and Technology from the source of the National Research, Development and Innovation Fund and from the János Bolyai Scholarship of the Hungarian Academy of Sciences.

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Benjamini, I., Fraczyk, M. & Kun, G. Expander spanning subgraphs with large girth. Isr. J. Math. 251, 156–172 (2022). https://doi.org/10.1007/s11856-022-2446-8

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  • DOI: https://doi.org/10.1007/s11856-022-2446-8

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