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 has 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. In this paper we prove the middle levels conjecture. In fact, we construct 2 Ω(n) different Hamilton cycles in the middle layer graph, which is best possible.

Construction of 2-factors in the middle layer of the discrete cube

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. In this work we present an inductive construction of a large family of 2-factors in the middle layer graph for all n?1. We also investigate how the choice of certain parameters used in the construction affects the number and lengths of the cycles in the resulting 2-factor.

Probabilistic One-Player Ramsey Games via Deterministic Two-Player Games

Consider the following probabilistic one-player game: The board is a graph with

Bipartite Kneser Graphs are Hamiltonian

n

A minimum-change version of the Chung–Feller theorem for Dyck paths

vertices, which initially contains no edges. In each step, a new edge is drawn uniformly at random from all nonedges and is presented to the player, henceforth called Painter. Painter must assign one of

Scheduling with few changes

r

Coloring random graphs online without creating monochromatic subgraphs

available colors to each edge immediately, where

Trimming and Gluing Gray Codes.

r\geq 2

An algorithmic framework for tool switching problems with multiple objectives

is a fixed integer. The game is over as soon as a monochromatic copy of some fixed graph

On universal partial words

F