A Combinatorial Proof of Kneser’s Conjecture

Authors:
Jiří Matoušek
Affiliations:
Charles University, Department of Applied Mathematics, Czech Republic and Institut für Informatik, ETH Zentrum, Zürich, Switzerland
Venue:
Combinatorica
Year:
2004

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Abstract

Kneser’s conjecture, first proved by Lovász in 1978, states that the graph with all k-element subsets of {1, 2, . . . , n} as vertices and with edges connecting disjoint sets has chromatic number n−2k+2. We derive this result from Tucker’s combinatorial lemma on labeling the vertices of special triangulations of the octahedral ball. By specializing a proof of Tucker’s lemma, we obtain self-contained purely combinatorial proof of Kneser’s conjecture.