Hamilton cycles in digraphs of unitary matrices

A set S subset of V is called a q(+)-set (q(-)-set, respectively) if S has at least two vertices and, for every u is an element of S, there exists nu is an element of S, nu not equal u such that N+(u) boolean AND N+(nu) not equal 0 (N-(u) boolean AND N-(nu) not equal 0, respectively). A digraph D is called s-quadrangular if, for every q+-set S, we have vertical bar U{N+(u) boolean AND N+(nu) : u not equal v, u, nu is an element of S}vertical bar >= vertical bar S vertical bar and, for every q(-)-set S, we have vertical bar U {N- (u) boolean AND N- (nu) : u, nu is an element of S}vertical bar >= vertical bar S vertical bar. We conjecture that every strong s-quadrangular digraph has a Hamilton cycle and provide some support for this conjecture. (c) 2006 Elsevier B.V. All rights reserved.