The restricted arc-connectivity λ′ of a strongly connected digraph G is the minimum cardinality of an arc cut F in G such that every strongly connected component of G−F contains at least two vertices. This paper shows that for a d-regular strongly connected digraph with order n and diameter k ≥ 4, if λ′ exists, then λ′(G) ≥ min { (n − dk−1)(d− 1) dk−1 + d− 2 , 2d− 2 } As consequences, the restr...

A digraph obtained by replacing each edge of a complete n-partite (n 2:: 2) graph by an arc or a pair of mutually opposite arcs with the same end vertices is called a semicomplete n-partite digraph or semicomplete multipartite digraph (abbreviated to SMD). In this paper we show the following result for a semicomplete multipartite digraph of order p with the partite sets VI, 112, ... , Vn. Let r...

A cellular embedding of an Eulerian digraph D into a closed surface is said to be directed if the boundary of each face is a directed closed walk in D. The directed genus polynomial of an Eulerian digraph D is the polynomial ΓD(x) = ∑ h≥0 gh(D)x h where gh(D) is the number of directed embeddings into the orientable surface Sh, of genus h, for h = 0, 1, . . . . The sequence {gh(D)|h ≥ 0}, which ...

A characterization of the adjacency matrix of the line digraph of a regular digraph is given. Some corollaries are observed.

Let D be a strongly k−connected digraph of order n ≥ 2. We prove that for every l ≥ n 2k the power Dl of D is Hamiltonian. Moreover, for any n > 2k ≥ 2 we exhibit strongly k-connected digraphs D of order n such that Dl−1 is non-Hamiltonian for l = d n 2ke. We use standard terminology, unless otherwise stated. A digraph D = (V, A) of order n ≥ 2 is said to be strongly q-arc Hamiltonian if for an...

In 1985, Mader conjectured the existence of a function f such that every digraph with minimum out-degree at least f(k) contains a subdivision of the transitive tournament of order k. This conjecture is still completely open, as the existence of f(5) remains unknown. In this paper, we show that if D is an oriented path, or an in-arborescence (i.e., a tree with all edges oriented towards the root...

From the theory of Ho0man polynomial, it is known that the adjacency matrix A of a strongly connected regular digraph of order n satis3es certain polynomial equation AP(A)=Jn, where l is a nonnegative integer, P(x) is a polynomial with rational coe5cients, and Jn is the n×n matrix of all ones. In this paper we present some su5cient conditions, in terms of the coe5cients of P(x), to ensure that ...

An almost Moore digraph G of degree d > 1, diameter k > 1 is a diregular digraph with the number of vertices is one less than the Moore bound. If G is an almost Moore digraph, then for each vertex u ∈ V (G) there exists a vertex v ∈ V (G), called repeat of u and denoted by r(u) = v, Such that there are two walks of lenght ≤ k from u to v. The smallest positive integer p such that the compositio...

The Directed Maximum Leaf Out-Branching problem is to find an out-branching (i.e. a rooted oriented spanning tree) in a given digraph with the maximum number of leaves. In this paper, we obtain two combinatorial results on the number of leaves in out-branchings. We show that – every strongly connected n-vertex digraph D with minimum indegree at least 3 has an out-branching with at least (n/4) −...

Let D be a digraph. Its reverse digraph, D−1, is obtained by reversing all arcs of D. We show that the domination numbers of D and D−1 can be different if D is a Cayley digraph. The smallest groups admitting Cayley digraphs with this property are the alternating group A4 and the dihedral group D6, both on 12 elements. Then, for each n ≥ 6 we find a Cayley digraph D on the dihedral group Dn such...