In this paper we investigate the following generalization of transitivity: A digraph D is (m, n )-transitive whenever there is a path of length m from x to y there is a subset of n + I vertices of these m +I vertices which contain a path of length n from x toy. Here we study various properties of (m, n )-transitive digraphs. In particular, (m, I)-transitive tournaments are characterized. Their ...

The pattern of a matrix M is a (0,1)-matrix which replaces all non-zero entries of M with a 1. There are several contexts in which studying the patterns of orthogonal matrices can be useful. One necessary condition for a matrix to be orthogonal is a property known as combinatorial orthogonality. If the adjacency matrix of a directed graph forms a pattern of a combinatorially orthogonal matrix, ...

A digraph D = (V, A) is arc-traceable if for each arc xy in A, xy lies on a directed path containing all the vertices of V , i.e., hamiltonian path. We prove a conjecture of Quintas [7]: if D is arc-traceable, then the condensation of D is a directed path. We show that the converse of this conjecture is false by providing an example of an upset tournament which is not arc-traceable. We then giv...

For a tournament T , let ν3(T ) denote the maximum number of pairwise arc-disjoint triangles in T . Let ν3(n) denote the minimum of ν3(T ) ranging over all regular tournaments with n vertices (n odd). It is conjectured that ν3(n) = (1 + on(1))n /9 and proved that n 11.43 (1− on(1)) ≤ ν3(n) ≤ n 9 (1 + on(1)) improving upon the best known upper bound and lower bound. The result is generalized to ...