Multicoloured Hamilton cycles in random graphs; an anti-Ramsey threshold

Abstract

Let the edges of a graph $G$ be coloured so that no colour is used more than $k$ times. We refer to this as a $k$-bounded colouring. We say that a subset of the edges of $G$ is multicoloured if each edge is of a different colour. We say that the colouring is $\cal H$-good, if a multicoloured Hamilton cycle exists i.e., one with a multicoloured edge-set. Let ${\cal AR}_k$ = $\{G :$ every $k$-bounded colouring of $G$ is $\cal H$-good$\}$. We establish the threshold for the random graph $G_{n,m}$ to be in ${\cal AR}_k$.