Note on alternating directed cycles

The problem of the existence of an alternating simple dicycle in a 2-arc-coloured digraph is considered. This is a generalization of the alternating cycle problem in 2-edgecoloured graphs (proved to be polynomial time solvable) and the even dicycle problem (the complexity is not known yet). We prove that the alternating dicycle problem is NPcomplete. Let f(n) (g(n), resp.) be the minimum integer such that if every monochromatic indegree and outdegree in a strongly connected 2-arc-coloured digraph (any 2-arccoloured digraph, resp.) D is at least f(n) (g(n), resp.), then D has an alternating simple dicycle. We show that f(n) = Θ(log n) and g(n) = Θ(log n).