Advertisement

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
Article

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.

Keywords

Chromatic number Circular chromatic number Kneser graphs 

Notes

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.

References

  1. Alon N, Frankl P, Lovász LL (1986) The chromatic number of Kneser hypergraphs. Trans Am Math Soc 298:359–370MathSciNetCrossRefzbMATHGoogle Scholar
  2. Bárány I (1978) A short of Kneser’s conjecture. J Combin Theory A 25:325–326CrossRefzbMATHGoogle Scholar
  3. Chang GJ, Liu DD-F, Zhu X (2013) A short proof for Chen’s Alternative Kneser Coloring Lemma. J Combin Theory A 120:159–163MathSciNetCrossRefzbMATHGoogle Scholar
  4. Chen P-A (2011) A new coloring theorem of Kneser graphs. J Combin Theory A 118(3):1062–1071MathSciNetCrossRefzbMATHGoogle Scholar
  5. Fan K (1952) A generalization of Tucker’s combinatorial lemma with topological applications. Ann Math 2(56):431–437CrossRefzbMATHGoogle Scholar
  6. Freund RM, Todd MJ (1981) A constructive proof of Tucker’s combinatorial lemma. J Combin Theory A 30:321–325MathSciNetCrossRefzbMATHGoogle Scholar
  7. Greene J (2002) A new short proof of Kneser’s conjecture. Am Math Monthly 109:918–920MathSciNetCrossRefzbMATHGoogle Scholar
  8. Hajiabolhassan H, Zhu X (2003) Circular chromatic number of Kneser graphs. J Combin Theory B 88(2):299–303MathSciNetCrossRefzbMATHGoogle Scholar
  9. Hajiabolhassan H, Taherkhani A (2010) Graph powers and graph homomorphisms. Electron J Combin 17(1):R17MathSciNetzbMATHGoogle Scholar
  10. Johnson A, Holroyd FC, Stahl S (1997) Multichromatic numbers, star chromatic numbers and Kneser graphs. J Graph Theory 26(3):137–145MathSciNetCrossRefzbMATHGoogle Scholar
  11. Kneser M (1955) Aufgabe 300. Jber Deutsch Math Verein 58:27Google Scholar
  12. Kriz I (1992) Equivalent cohomology and lower bounds for chromatic numbers. Trans Am Math Soc 333:567–577MathSciNetCrossRefzbMATHGoogle Scholar
  13. Kriz I (2000) A corretion to “Equivalent cohomology and lower bounds for chromatic numbers. Trans Am Math Soc 352:1951–1952MathSciNetCrossRefGoogle Scholar
  14. Lih K-W, Liu DD-F (2002) Circular chromatic numbers of some reduced Kneser graphs. J Graph Theory 41:62–68MathSciNetCrossRefzbMATHGoogle Scholar
  15. Lovász L (1978) Kneser’s conjecture, chromatic number, and homotopy. J Combin Theory A 25(3):319–324CrossRefzbMATHGoogle Scholar
  16. Matoušek J (2003) Using the Borsuk–Ulam theorem: lectures on topological methods in combinatorics and geometry. Springer, BerlinzbMATHGoogle Scholar
  17. Matoušek J (2004) A combinatorial proof of Kneser’s conjecture. Combinatorica 24:163–170MathSciNetCrossRefzbMATHGoogle Scholar
  18. Meunier F (2005) A topological lower bound for the circular chromatic number of Schrijver graphs. J Graph Theory 49(4):257–261MathSciNetCrossRefzbMATHGoogle Scholar
  19. Prescott T, Su F (2005) A constructive proof of Ky Fan’s generalization of Tucker’s lemma. J Combin Theory A 111:257–265MathSciNetCrossRefzbMATHGoogle Scholar
  20. Sarkaria KS (1990) A generalized Kneser conjecture. J Combin Theory B 49:236–240MathSciNetCrossRefzbMATHGoogle Scholar
  21. Schrijver A (1978) Vertex-critical subgraphs of Kneser graphs. Nieuw Arch Wiskd III 26:454–461MathSciNetzbMATHGoogle Scholar
  22. Simonyi G, Tardos G (2006) Local chromatic number, Ky Fan’s theorem and circular colorings. Combinatorica 26(5):587–626MathSciNetCrossRefzbMATHGoogle Scholar
  23. Tucker AW (1946) Some topological properties of disk and sphere. In: Proceedings of the first Canadian Mathematical Congress, Montreal. University of Toronto Press, Toronto, pp 285–309Google Scholar
  24. Zhu X (2001) Circular chromatic number: a survey. Discrete Math 229(1–3):371–410MathSciNetCrossRefzbMATHGoogle Scholar
  25. Zhu X (2006) Recent developments in circular colouring of graphs. Topics in discrete mathematics. Algorithms and combinatorics, vol 26. Springer, Berlin, pp 497–550Google Scholar
  26. Zhu X (2012) Circular coloring and flow. Lecture noteGoogle Scholar
  27. Ziegler G (2002) Generalized Kneser coloring theorems with combinatorial proofs. Invent Math 147:671–691MathSciNetCrossRefzbMATHGoogle Scholar

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

Personalised recommendations