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

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 H-good, if a multicoloured Hamilton cycle exists i.e., one with a multicoloured edge-set. Let ARk = {G : every k-bounded colouring of G is H-good}. We establish the threshold for the random graph Gn,m to be in ARk. ∗Research carried out whilst visiting Carnegie Mellon University †Supported by NSF grant CCR-9225008