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Non-Bipartite K-Common Graphs

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Abstract

A graph H is k-common if the number of monochromatic copies of H in a k-edge-coloring of Kn is asymptotically minimized by a random coloring. For every k, we construct a connected non-bipartite k-common graph. This resolves a problem raised by Jagger, Štovíček and Thomason [20]. We also show that a graph H is k-common for every k if and only if H is Sidorenko and that H is locally k-common for every k if and only if H is locally Sidorenko.

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Acknowledgement

The authors would like to thank László Miklós Lovász for insightful comments on the graph limit notions discussed in this paper, and the two anonymous reviewers for preparing very detailed reports in an unusually fast way; the comments of the reviewers helped to substantially improve the presentation of our results.

Author information

Authors and Affiliations

  1. Faculty of Informatics, Masaryk University, Botanická 68A, 602 00, Brno, Czech Republic

    Daniel Král’

  2. Mathematics Institute DIMAP and Department of Computer Science, University of Warwick, Coventry, CV4 7AL, UK

    Daniel Král’

  3. Department of Mathematics and Statistics, University of Victoria, David Turpin Building A425 3800 Finnerty Road, Victoria, Canada, V8P 5C2

    Jonathan A. Noel

  4. Mathematics Institute and DIMAP, University of Warwick, Coventry, CV4 7AL, UK

    Jonathan A. Noel

  5. Department of Mathematics and Statistics, McGill University, Montreal, Canada

    Sergey Norin

  6. Department of Mathematics Faculty of Nuclear Sciences and Physical Engineering, Czech Technical University in Prague, Trojanova 13, 120 00, Prague, Czech Republic

    Jan Volec

  7. Faculty of Informatics, Masaryk University, Botanická 68A, 602 00, Brno, Czech Republic

    Jan Volec

  8. Department of Mathematics, Princeton University, Princeton, NJ, USA

    Fan Wei

  9. School of Mathematics, Institute for Advanced Study, Princeton, USA

    Fan Wei

Authors
  1. Daniel Král’
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  2. Jonathan A. Noel
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  3. Sergey Norin
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  4. Jan Volec
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  5. Fan Wei
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Corresponding author

Correspondence to Jan Volec.

Additional information

The work of the first, second and fourth authors has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No 648509). This publication reflects only its authors’ view; the European Research Council Executive Agency is not responsible for any use that may be made of the information it contains. The first and the fourth author were also supported by the MUNI Award in Science and Humanities of the Grant Agency of Masaryk University. The second author was also supported by the Leverhulme Trust Early Career Fellowship ECF-2018-534. The third author was supported by an NSERC Discovery grant. The fifth author was supported by IAS Founders’ Circle funding provided by Cynthia and Robert Hillas.

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Král’, D., Noel, J.A., Norin, S. et al. Non-Bipartite K-Common Graphs. Combinatorica 42, 87–114 (2022). https://doi.org/10.1007/s00493-020-4499-9

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  • DOI: https://doi.org/10.1007/s00493-020-4499-9

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