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Regular partitions of gentle graphs

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Abstract

Szemerédi's Regularity Lemma is a very useful tool of extremal combinatorics. Recently, several refinements of this seminal result were obtained for special, more structured classes of graphs. We survey these results in their rich combinatorial context. In particular, we stress the link to the theory of (structural) sparsity, which leads to alternative proofs, refinements and solutions of open problems. It is interesting to note that many of these classes present challenging problems. Nevertheless, from the point of view of regularity lemma type statements, they appear as ``gentle'' classes.

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Acknowledgement

The authors are indebted to O-joung Kwon, who pointed out an error in an earlier version of this paper.

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Authors and Affiliations

  1. Institut de recherche en informatique fondamentale, Paris, France

    Y. Jiang

  2. Computer Science Institute of Charles University (IUUK), Praha, Czech Republic

    J. Nešetřil & P. Ossona de Mendez

  3. Centre d’Analyse et de Mathématiques Sociales (CNRS, UMR 8557), Paris, France

    P. Ossona de Mendez

  4. University of Bremen, Bremen, Germany

    S. Siebertz

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  1. Y. Jiang
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  2. J. Nešetřil
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  3. P. Ossona de Mendez
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Correspondence to J. Nešetřil.

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Dedicated to Endre Szemerédi on the occasion of his eightieth birthday

This paper is part of a project that has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No. 810115 – Dynasnet).

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Jiang, Y., Nešetřil, J., Ossona de Mendez, P. et al. Regular partitions of gentle graphs. Acta Math. Hungar. 161, 719–755 (2020). https://doi.org/10.1007/s10474-020-01074-x

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