A sufficient condition for the existence of an anti-directed 2-factor in a directed graph

Let D be a directed graph with vertex set V , arc set A , and order n . The graph underlying D is the graph obtained from D by replacing each arc ( u , v ) ∈ A by an undirected edge { u , v } and then replacing each double edge by a single edge. An anti-directed (hamiltonian) cycle H in D is a (hamiltonian) cycle in the graph underlying D such that no pair of consecutive arcs in H form a directed path in D . An anti-directed 2-factor in D is a vertex-disjoint collection of anti-directed cycles in D that span V . It was proved in Busch et al. (submitted for publication) [3] that if the indegree and the outdegree of each vertex of D is greater than 9 16 n then D contains an anti-directed Hamilton cycle. In this paper we prove that given a directed graph D , the problem of determining whether D has an anti-directed 2-factor is NP-complete, and we use a proof technique similar to the one used in Busch et al. (submitted for publication) [3] to prove that if the indegree and the outdegree of each vertex of D is greater than 24 46 n then D contains an anti-directed 2-factor.