Multicoloured Hamilton Cycles

Abstract

The edges of the complete graph $K_n$ are coloured so that no colour appears more than $\lceil cn\rceil$ times, where $c < 1/32$ is a constant. We show that if $n$ is sufficiently large then there is a Hamiltonian cycle in which each edge is a different colour, thereby proving a 1986 conjecture of Hahn and Thomassen. We prove a similar result for the complete digraph with $c < 1/64$. We also show, by essentially the same technique, that if $t\geq 3$, $c