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A combinatorial proof for the circular chromatic number of Kneser graphs

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Abstract

Chen (J Combin Theory A 118(3):1062–1071, 2011) confirmed the Johnson–Holroyd–Stahl conjecture that the circular chromatic number of a Kneser graph is equal to its chromatic number. A shorter proof of this result was given by Chang et al. (J Combin Theory A 120:159–163, 2013). Both proofs were based on Fan’s lemma (Ann Math 56:431–437, 1952) in algebraic topology. In this article we give a further simplified proof of this result. Moreover, by specializing a constructive proof of Fan’s lemma by Prescott and Su (J Combin Theory A 111:257–265, 2005), our proof is self-contained and combinatorial.

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Acknowledgments

The authors would like to thank the two anonymous referees for their suggestions, which resulted in better presentation of this article. X. Zhu: Grant Numbers: NSF11171310 and ZJNSF Z6110786.

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Authors and Affiliations

  1. Department of Mathematics, California State University, Los Angeles, Los Angeles, USA

    Daphne Der-Fen Liu

  2. Department of Mathematics, Zhejiang Normal University, Jinhua, China

    Xuding Zhu

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  1. Daphne Der-Fen Liu
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  2. Xuding Zhu
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Correspondence to Daphne Der-Fen Liu.

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Liu, D.DF., Zhu, X. A combinatorial proof for the circular chromatic number of Kneser graphs. J Comb Optim 32, 765–774 (2016). https://doi.org/10.1007/s10878-015-9897-3

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  • DOI: https://doi.org/10.1007/s10878-015-9897-3

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