  A set $S\subseteq V$ is called an {\em $q^+$-set} ({\em $q^-$-set},
respectively) if $S$ has at least two vertices and, for every $u\in S$, there
exists $v\in S, v\neq u$ such that $N^+(u)\cap N^+(v)\neq \emptyset$
($N^-(u)\cap N^-(v)\neq \emptyset$, respectively). A digraph $D$ is called {\em
s-quadrangular} if, for every $q^+$-set $S$, we have $|\cup \{N^+(u)\cap
N^+(v): u\neq v, u,v\in S\}|\ge |S|$ and, for every $q^-$-set $S$, we have
$|\cup \{N^-(u)\cap N^-(v): u,v\in S)\}\ge |S|$. We conjecture that every
strong s-quadrangular digraph has a Hamilton cycle and provide some support for
this conjecture.
