In 1960 Ghouila-Houri extended Dirac’s theorem to directed graphs by proving that if D is a directed graph on n vertices with minimum out-degree and in-degree at least n/2, then D contains a directed Hamiltonian cycle. For directed graphs one may ask for other orientations of a Hamiltonian cycle and in 1980 Grant initiated the problem of determining minimum degree conditions for a directed grap...

A directed cycle double cover of a graph G is a family of cycles of G, each provided with an orientation, such that every edge of G is covered by exactly two oppositely directed cycles. Explicit obstructions to the existence of a directed cycle double cover in a graph are bridges. Jaeger [4] conjectured that bridges are actually the only obstructions. One of the difficulties in proving the Jaeg...

When we represent real-world systems as networks, the directions of links often convey valuable information. Finding module structures that respect link directions is one of the most important tasks for analysing directed networks. Although many notions of a directed module have been proposed, no consensus has been reached. This lack of consensus results partly because there might exist distinc...

A 2k-dicycle is a 2k-cycle of a directed bipartite graph and a 2k-dicycle system of order (m,n) is a triple (X, Y,D), where D is a collection of edge disjoint 2k-dicycles which partitions the edge set of the complete directed bipartite graph Dm,n with parts X and Y . A partial 2k-dicycle system of order (s, t) is a triple (S, T, P ), where P is a collection of edge disjoint 2k-dicycles of Ds,t ...

Given a directed graph G and a parameter k, the Long Directed Cycle (LDC) problem asks whether G contains a simple cycle on at least k vertices, while the k-Path problems asks whether G contains a simple path on exactly k vertices. Given a deterministic (randomized) algorithm for k-Path as a black box, which runs in time t(G, k), we prove that LDC can be solved in deterministic time O(max{t(G, ...

it is known that the directed cycle of order $n$ uniquely achieves the minimum spectral radius among all strongly connected digraphs of order $nge 3$. in this paper, among others, we determine the digraphs which achieve the second, the third and the fourth minimum spectral radii respectively among strongly connected digraphs of order $nge 4$.

A directed m-cycle system is a collection of directed edges of the form {(XbX2), (X2' X3)(X3, X4), ... , (Xm-l, xm), (xm' xd}, where Xl, X2, X3, ... , Xm are distinct. A partial directed m-cycle system of order n is a pair (8, C), where C is a collection of edge disjoint m-cycles of the complete directed graph Dn with vertex set S. If the cycles in C partition the edge set of Dn we have the def...

If each arc in a strongly connected directed graph of minimum in-degree and outdegree at least 3 is assigned a weight 0 or 1, then the resultingweighted directed graph has a directed cycle of even total weight. This provesa conjecture made by L. Lovasz in 1975 and has applications to colour-criticalhypergraphs, sign-nonsingular matrices, and permanents of matrices. MATHEMATICAL ...

The problem of decomposing networks into modules (or clusters) has gained much attention in recent years, as it can account for a coarsegrained description of complex systems, often revealing functional subunits of these systems. A variety of module detection algorithms have been proposed, mostly oriented towards finding hard partitionings of undirected networks. Despite the increasing number o...