Journal of Combinatorial Optimization

, Volume 32, Issue 3, pp 765–774 | Cite as

A combinatorial proof for the circular chromatic number of Kneser graphs

  • Daphne Der-Fen LiuEmail author
  • Xuding Zhu


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.


Chromatic number Circular chromatic number Kneser graphs 



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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Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.Department of MathematicsCalifornia State University, Los AngelesLos AngelesUSA
  2. 2.Department of MathematicsZhejiang Normal UniversityJinhuaChina

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