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