Bipartite Kneser graphs are Hamiltonian

Graph Theory Seminar
Thursday, April 9, 2015 - 12:05pm
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
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).