Note on alternating directed cycles

Abstract

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-edge-coloured graphs and the even dicycle problem (both are polynomial time solvable). We prove that the alternating dicycle problem is -complete. Let ƒ(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-arc-coloured digraph, resp.) D is at least f(n) (g(n), resp.), then D has an alternating simple dicycle. We show that ƒ(n) = Θ(log n) and g(n) = Θ(log n).