A combinatorial proof for the circular chromatic number of Kneser graphs
- 207 Downloads
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.
KeywordsChromatic 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.
- Kneser M (1955) Aufgabe 300. Jber Deutsch Math Verein 58:27Google Scholar
- 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
- 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
- Zhu X (2012) Circular coloring and flow. Lecture noteGoogle Scholar