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

Abstract

Let D be a directed graph with vertex set V and order n. 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 [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 [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.