J. Robert Johnson

Universal cycles for permutations

A universal cycle for permutations is a word of length n! such that each of the n! possible relative orders of n distinct integers occurs as a cyclic interval of the word. We show how to construct such a universal cycle in which only n+1 distinct integers are used. This is best possible and proves a conjecture of Chung, Diaconis and Graham.

Explicit 2-Factorisations of the Odd Graph

In this note we show how 1-factors in the middle two layers of the discrete cube can be used to construct 2-factors in the Odd graph (the Kneser graph of (k − 1)-sets from a (2k − 1)-set). In particular, we use the lexical matchings of Kierstead and Trotter, and the modular matchings of Duffus, Kierstead and Snevily, to give explicit constructions of two different 2-factorisations of the Odd graph.

Multicolour Ramsey Numbers of Odd Cycles

We show that for any positive integer

Saturated Subgraphs of the Hypercube

r

Turán and ramsey properties of subcube intersection graphs

there exists an integer

A Disproof of the Fon-der-Flaass Conjecture

k

Random majority percolation

and a

Long cycles in the middle two layers of the discrete cube

k

Vertex Turán problems in the hypercube

-colouring of the edges of

An inductive construction for Hamilton cycles in Kneser graphs

K_{2^{k}+1}