Bipartite Kneser graphs are Hamiltonian

Graph Theory Seminar
Thursday, April 9, 2015 - 12:05
1 hour (actually 50 minutes)
Skiles 005
School of Mathematics, Georgia Tech and ETH Zurich
For integers k>=1 and n>=2k+1, the bipartite Kneser graph H(n,k) is defined as the graph that has as vertices all k-element and all (n-k)-element subsets of {1,2,...,n}, with an edge between any two vertices (=sets) where one is a subset of the other. It has long been conjectured that all bipartite Kneser graphs have a Hamilton cycle. The special case of this conjecture concerning the Hamiltonicity of the graph H(2k+1,k) became known as the 'middle levels conjecture' or 'revolving door conjecture', and has attracted particular attention over the last 30 years. One of the motivations for tackling these problems is an even more general conjecture due to Lovasz, which asserts that in fact every connected vertex-transitive graph (as e.g. H(n,k)) has a Hamilton cycle (apart from five exceptional graphs). Last week I presented a (rather technical) proof of the middle levels conjecture. In this talk I present a simple and short proof that all bipartite Kneser graphs H(n,k) have a Hamilton cycle (assuming that H(2k+1,k) has one). No prior knowledge will be assumed for this talk (having attended the first talk is not a prerequisite). This is joint work with Pascal Su (ETH Zurich).