Hamilton cycles in digraphs of unitary matrices

A set S⊆VS⊆V is called a q+q+-set   (q-q--set, respectively) if S   has at least two vertices and, for every u∈Su∈S, there exists v∈S,v≠uv∈S,v≠u such that N+(u)∩N+(v)≠∅N+(u)∩N+(v)≠∅ (N-(u)∩N-(v)≠∅N-(u)∩N-(v)≠∅, respectively). A digraph D is called s-quadrangular   if, for every q+q+-set S  , we have |∪{N+(u)∩N+(v):u≠v,u,v∈S}|⩾|S||∪{N+(u)∩N+(v):u≠v,u,v∈S}|⩾|S| and, for every q-q--set S  , we have |∪{N-(u)∩N-(v):u,v∈S)}⩾|S||∪{N-(u)∩N-(v):u,v∈S)}⩾|S|. We conjecture that every strong ss-quadrangular digraph has a Hamilton cycle and provide some support for this conjecture.