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Abstra t Let the edges of a graph G be oloured so that no olour is used more than k times. We refer to this as a k-bounded olouring. We say that a subset of the edges of G is multi oloured if ea h edge is of a di erent olour. We say that the olouring is H-good, if a multi oloured Hamilton y le exists i.e., one with a multi oloured edgeset. Let ARk = fG : every k-bounded olouring of G is H-goodg. We establish the threshold for the random graph Gn;m to be in ARk.