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

Date and Time: Thursday, November 01, 2018, 12:15 pm

Duration: 30 minutes

Location: OAT S15/S16/S17

Speaker: Jerri Nummenpalo

A simple proof of the middle levels theorem

For a positive integer k, the middle levels graph M_k is the bipartite graph that appears between the two middle levels of a Boolean hypercube of dimension 2k+1. In other words, the vertices of M_k are all bitstrings of length 2k+1 with either k or k+1 bits equal to 1, and two vertices are adjacent if they differ in exactly one bit. A conjecture, that became known as the middle levels conjecture, asserts that M_k is Hamiltonian, i.e., that there is a cycle that covers the whole graph. The conjecture seems to have first appeared in the 1980s in papers by Havel and by Buck and Wiedemann, though some unclarity of its origin remains, and it has also been attributed to Erdős, Trotter and Dejter.

The middle levels conjecture was resolved recently on the positive by Mütze. His proof is a technical, 37-page paper and our result is a much shorter, simpler and more accessible proof. In this talk I will go through the main ideas of our proof and show a connection between the lexical matchings in M_k, plane trees and Catalan numbers.

Based on joint work with Petr Gregor and Torsten Mütze.

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