Mittagsseminar (in cooperation with A. Steger, D. Steurer and B. Sudakov)

Date and Time: Thursday, April 17, 2014, 12:15 pm

Duration: 30 minutes

Location: OAT S15/S16/S17

Speaker: Torsten Mütze

Proof of the middle levels conjecture

Define the middle layer graph as the graph whose vertex set consists of all bitstrings of length 2n+1 that have exactly n or n+1 entries equal to 1, with an edge between any two vertices for which the corresponding bitstrings differ in exactly one bit. The middle levels conjecture asserts that this graph contains a Hamilton cycle for every n>=1. This conjecture originated probably with Havel, Buck and Wiedemann, but has also been attributed to Dejter, Erdős, Trotter and various others, and despite considerable efforts it remained open during the last 30 years. One of the motivations for tackling this conjecture is an even more general conjecture due to Lovász, which asserts that every connected vertex-transitive graph (as e.g. the middle layer graph) contains a Hamilton path, and apart from five exceptional graphs, even a Hamilton cycle.
In this talk I present a proof of the middle levels conjecture. In fact, I show that the middle layer graph contains 2^{2^{\Omega(n)}} different Hamilton cycles, which is best possible.

http://www.openproblemgarden.org/op/middle_levels_problem
http://www.math.uiuc.edu/~west/openp/revolving.html

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