Antidirected Hamiltonian Paths and Cycles of Digraphs with $\alpha _{2}$-Stable Number 2

Abstract

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 $\alpha _{2}(D) = \max \{|W|: W \subseteq V(D)$ and D[W] has no 2-cycle $\}$ be the $\alpha _{2}(D)$-stable number. In this paper, we show that every weakly connected digraph D with $\alpha _{2}(D)=2$ has an antidirected Hamiltonian path. Secondly, we determine two families of well-characterized strongly connected digraphs ${\mathcal {H}}$ and ${\mathcal {M}}$ such that for any strongly connected digraph $D\in {\mathcal {H}}\cup {\mathcal {M}}$ which has no antidirected Hamiltonian cycle. And finally, we further prove that every strongly connected digraph D with $\alpha _{2}(D)=2$ has an antidirected Hamiltonian cycle if and only if |V(D)| is even and $D\not \in {\mathcal {H}}\cup {\mathcal {M}}$.