Bipartite Kneser Graphs are Hamiltonian

Abstract

For integers k ≥1 and n≥2k+1 the Kneser graph K(n;k) has as vertices all k-element subsets of [n]:={1;2;:::;n} and an edge between any two vertices (=sets) that are disjoint. The bipartite Kneser graph H(n,k) has as vertices all k-element and (n—k)-element subsets of [n] and an edge between any two vertices where one is a subset of the other. It has long been conjectured that all Kneser graphs and bipartite Kneser graphs except the Petersen graph K(5, 2) have a Hamilton cycle. The main contribution of this paper is proving this conjecture for bipartite Kneser graphs H(n,k). We also establish the existence of cycles that visit almost all vertices in Kneser graphs K(n,k) when n=2k+o(k), generalizing and improving upon previous results on this problem.

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Correspondence to Torsten Mütze.

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An extended abstract of this work has appeared in the proceedings of the European Conference on Combinatorics, Graph Theory and Applications (Eurocomb) 2015.

The author was supported by a fellowship of the Swiss National Science Foundation. This work was completed when the author was with the School of Mathematics at Georgia Institute of Technology, 30332 Atlanta GA, USA.

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Mütze, T., Su, P. Bipartite Kneser Graphs are Hamiltonian. Combinatorica 37, 1207–1219 (2017). https://doi.org/10.1007/s00493-016-3434-6

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Mathematics Subject Classification (2000)

  • 05C45