It is well known that every 2-edge-connected graph can be oriented so that the resulting digraph is strongly connected. Here we study the problem of orienting a connected graph with cut edges in order to maximize the number of ordered vertex pairs (x,y) such that there is a directed path from x to y. After transforming this problem, we prove a key theorem about the transformed problem that allo...

We show that a digraph which contains a directed 2-factor and has minimum in-degree and out-degree at least four has two non-isomorphic directed 2-factors. As a corollary we deduce that every graph which contains a 2factor and has minimum degree at least eight has two non-isomorphic 2factors. In addition we construct: an infinite family of strongly connected 3-diregular digraphs with the proper...

Let G be a strongly connected digraph, and dG(vi,vj) denote the distance from vertex vi to vj defined as length of shortest directed path in G. The sum between vertices is sdG(vi,vj)=dG(vi,vj)+dG(vj,vi). matrix n×n SD(G)=(sdG(vi,vj))vi,vj∈V(G). For vi∈V(G), transmission G, denoted by STG(vi) or STi, row SD(G) corresponding vi. ST(G)=diag(ST1,ST2,…,STn) diagonal with transmissions zeroes elsewhe...

We develop a combinatorial polynomial-time algorithm to make a (k − 1)-connected digraph k-connected by adding a minimum number of new edges. © 2008 Elsevier B.V. All rights reserved.

A k-geodetic digraph G is a digraph in which, for every pair of vertices u and v (not necessarily distinct), there is at most one walk of length ≤ k from u to v. If the diameter of G is k, we say that G is strongly geodetic. Let N(d, k) be the smallest possible order for a k-geodetic digraph of minimum out-degree d, then N(d, k) ≥ 1 + d+ d2 + . . .+ dk = M(d, k), where M(d, k) is the Moore boun...

We consider the Erd\H{o}s-P\'osa property for immersions and topological minors in tournaments. prove that every simple digraph $H$, $k\in \mathbb{N}$, tournament $T$, following statements hold: (i) If $T$ one cannot find $k$ arc-disjoint immersion copies of then there exists a set $\mathcal{O}_H(k^3)$ arcs intersects all $H$ $T$. (ii) vertex-disjoint minor $\mathcal{O}_H(k\log k)$ vertices Thi...

It is shown that for each r 3, a random r-regular graph on 2n vertices is equivalent in a certain sense to a set of r randomly chosen disjoint perfect matchings of the 2n vertices, as n ! 1. This equivalence of two sequences of probabilistic spaces, called contiguity, occurs when all events almost sure in one sequence of spaces are almost sure in the other, and vice versa. The corresponding sta...

Let D be a simple digraph without loops or digons (i.e. if (u, v) ∈ E(D), then (v, u) 6∈ E(D)). For any v ∈ V (D) let N1(v) be the set of all vertices at out-distance 1 from v and let N2(v) be the set of all vertices at out-distance 2. We provide sufficient conditions under which there must exist some v ∈ V (D) such that |N1(v)| ≤ |N2(v)|, as well as examine properties of a minimal graph which ...