Vertex-oriented Hamilton cycles in directed graphs

Let D be a directed graph of order n. An anti-directed Hamilton cycle H in D is a Hamilton cycle in the graph underlying D such that no pair of consecutive arcs in H form a directed path in D. We prove that if D is a directed graph with even order n and if the indegree and the outdegree of each vertex of D is at least 2/3n then D contains an anti-directed Hamilton cycle. This improves a bound of Grant [7]. Let V(D) = P boolean OR Q be a partition of V(D). A (P,Q) vertex-oriented Hamilton cycle in D is a Hamilton cycle H in the graph underlying D such that for each v is an element of P, consecutive arcs of H incident on v do not form a directed path in D, and, for each v is an element of Q,consecutive arcs of H incident on v form a directed path in D. We give sufficient conditions for the existence of a (P,Q) vertex-oriented Hamilton cycle in D for the cases when |P| >= 2/3n and when 1/3n <= |P| <= 2/3n. This sharpens a bound given by Badheka et al. in [1].