Antidirected Hamiltonian Paths and Cycles of Digraphs with α2-Stable Number 2.

Let D be a digraph with vertex set V(D) and arc set A(D). An antidirected Hamiltonian path (resp. cycle) of D is a Hamiltonian path (resp. cycle) in which consecutive arcs have opposite directions and each arc of D occurs exactly once. Let α 2 (D) = max { | W | : W ⊆ V (D) and D[W] has no 2-cycle } be the α 2 (D) -stable number. In this paper, we show that every weakly connected digraph D with α 2 (D) = 2 has an antidirected Hamiltonian path. Secondly, we determine two families of well-characterized strongly connected digraphs H and M such that for any strongly connected digraph D ∈ H ∪ M which has no antidirected Hamiltonian cycle. And finally, we further prove that every strongly connected digraph D with α 2 (D) = 2 has an antidirected Hamiltonian cycle if and only if |V(D)| is even and D ∉ H ∪ M . [ABSTRACT FROM AUTHOR]