On non-even digraphs and symplectic pairs

Abstract

A digraph $D$ is called {\bf noneven} if it is possible to assign weights of 0,1 to its arcs so that $D$ contains no cycle of even weight. A noneven digraph $D$ corresponds to one or more nonsingular sign patterns. Given an $n \times n$ sign pattern $H$, a {\bf symplectic pair} in $Q(H)$ is a pair of matrices $(A,D)$ such that $A \in Q(H)$, $D \in Q(H)$, and $A^T D = I$. An unweighted digraph $D$ allows a matrix property $P$ if at least one of the sign patterns whose digraph is $D$ allows $P$. Thomassen characterized the noneven, 2-connected symmetric digraphs (i.e., digraphs for which the existence of arc $(u,v)$ implies the existence of arc $(v ,u))$. In the first part of our paper, we recall this characterization and use it to determine which strong symmetric digraphs allow symplectic pairs. A digraph $D$ is called {\bf semi-complete} if, for each pair of distinct vertices $(u,v)$, at least one of the arcs digraph. In the second part of our paper, we fill a gap in these two characterizations and present and prove correct versions of the main theorems involved. We then pr oceed to determine which digraphs from these classes allow symplectic pairs. $(u,v)$ and $(v,u)$ exists in $D$. Thomassen presented a characterization of two classes of strong, noneven digraphs: the semi-complete class and the class of digraphs for which each vertex has total degree which exceeds or equals the size of the digraph. In the second part of our paper, we fill a gap in these two characterizations and present and prove correct versions of the main theorems involved. We then p oceed to determine which digraphs from these classes allow symplectic pairs.