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On 4-chromatic Schrijver graphs: their structure, non-3-colorability, and critical edges

A Story with Drums$

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Abstract

We investigate 4-chromatic Schrijver graphs from various points of view and color-critical edges in Schrijver graphs in general. In particular, we present the following results.

  • We give an elementary proof for the non-3-colorability of 4-chromatic Schrijver graphs thus providing such a proof also for 4-chromatic Kneser graphs.

  • We show that only certain types of edges of Schrijver graphs can be color-critical and prove that many of those are indeed color-critical, though in general not all of them are. In the 4-chromatic case these results give a complete characterization of the color-critical edges.

  • We show that a spanning subgraph of 4-chromatic Schrijver graphs quadrangulates the Klein bottle, while another spanning subgraph quadrangulates the projective plane. The latter is a special case of a result by Kaiser and Stehlík.

  • We show that (apart from two cases of small parameters) the subgraphs we present that quadrangulate the Klein bottle are edge-color-critical. The analogous result for the subgraphs quadrangulating the projective plane is an immediate consequence of earlier results by Gimbel and Thomassen and was already noted by Kaiser and Stehlík.

Our proof of non-3-colorability of 4-chromatic Schrijver graphs is based on a complete description of their structure that we present as well. This was already also given (in somewhat different terms) by Braun and even earlier in an unpublished manuscript by Li.

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Acknowledgements

We thank Tomáš Kaiser and Matěj Stehlík for a useful conversation and in particular for drawing our attention to the paper [16]. We also thank an anonymous referee for a very thorough reading of our manuscript and for giving valuable suggestions.

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Authors and Affiliations

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

    G. Simonyi & G. Tardos

  2. Department of Computer Science and Information Theory, Budapest University of Technology and Economics, Budapest, Hungary

    G. Simonyi

  3. Department of Mathematics, Central European University, Budapest, Hungary

    G. Tardos

Authors
  1. G. Simonyi
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  2. G. Tardos
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Correspondence to G. Simonyi.

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Dedicated to Endre Szemerédi on the occasion of his 80th birthday

$This subtitle, on the one hand, refers to Definition 4 in Section 2, while, on the other hand, to the joke said by Endre Szemerédi in the beginning of his Abel lecture (which is available on youtube).

Research partially supported by the National Research, Development and Innovation Office (NKFIH) grants K-116769, K-120706, K-132696 and by the National Research, Development and Innovation Fund (TUDFO/51757/2019-ITM, Thematic Excellence Program).

Research partially supported by the Cryptography “Lendület” project of the Hungarian Academy of Sciences and by the National Research, Development and Innovation Office (NKFIH) grants K-116769, K-132696 and KKP-133864, by the ERC Synergy Grant “Dynasnet” No. 810115 and by the Ministry of Education and Science of the Russian Federation in the framework of MegaGrant No. 075-15-2019-1926.

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Simonyi, G., Tardos, G. On 4-chromatic Schrijver graphs: their structure, non-3-colorability, and critical edges. Acta Math. Hungar. 161, 583–617 (2020). https://doi.org/10.1007/s10474-020-01080-z

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  • DOI: https://doi.org/10.1007/s10474-020-01080-z

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