A Note on Vertex Pancyclic Oriented Graphs

Let D be an oriented graph of order n 9, minimum degree n \Gamma 2, such for choice of distinct vertices x and y, either xy 2 E(D) or d + (x)+d \Gamma (y) n \Gamma 3. Song (J. Graph Theory 18 (1994), 461--468) proved that D is pancyclic. In this note, we give a short proof, based on Song's result, that D is in fact vertex pancyclic. This also generalizes a result of Jackson (J. Graph Theory 5 (1981), 147--157) for the existence of a hamiltonian cycle in oriented graphs. 1 Terminology and Introduction An oriented graph is a digraph without loops, multiple arcs, or cycles of length 2. The vertex set and the arc set of a digraph D are denoted by V (D) and E(D), respectively. If xy is an arc of D, then we say that x dominates y. More generally, for two disjoint subdigraphs A and B of D, if every vertex of A dominates every vertex of B, then we say that A dominates B, denoted by A ! B. In addition, we sometimes use the notation A ) B to denote that there is no arc from B to A. The ..