Digraphs Of Diameter Research Articles

K‐quasi‐transitive digraphs of large diameter

AbstractGiven an integer with , a digraph is ‐quasi‐transitive if for every ‐directed path of length in , we have or (or both). In this study, we prove that if is an odd integer, , then every strong ‐quasi‐transitive digraph of diameter at least admits a partition of its vertex set such that is Hamiltonian, and both and are semicomplete bipartite, when is bipartite, or semicomplete, otherwise. As a consequence, for an odd integer , it is easy to prove that every non‐bipartite strong ‐quasi‐transitive digraph with diameter at least has a Hamiltonian path.

K-quasi-transitive digraphs of large diameter

Let k be an integer, k≥2. A digraph D=(V,A) is k-quasi-transitive if for every pair of vertices u,v∈V, the existence of a directed path of length k from u to v implies the existence of the arc (u, v) or (v, u) in A(D). Under this definition, quasi-transitive digraphs are 2-quasi-transitive digraphs.A recursive characterization (the so-called Canonical Decomposition) is known for quasi-transitive digraphs, but no characterization is known for k-quasi-transitive digraphs in the general case. Recently, Wang and Zhang proved that if k is an even integer, then a k-quasi-transitive digraph of diameter at least k+2 admits a partition of its vertex set into two parts, each of them inducing a semicomplete digraph. In this work, we will present an analogous result for the case when k is an odd integer and discuss some of its consequences and future lines of research.

Large butterfly Cayley graphs and digraphs

We present families of large undirected and directed Cayley graphs whose construction is related to butterfly networks. One approach yields, for every large k and for values of d taken from a large interval, the largest known Cayley graphs and digraphs of diameter k and degree d. Another method yields, for sufficiently large k and infinitely many values of d, Cayley graphs and digraphs of diameter k and degree d whose order is exponentially larger in k than any previously constructed. In the directed case, these are within a linear factor in k of the Moore bound.

Diameter and maximum degree in Eulerian digraphs

In this paper we consider upper bounds on the diameter of Eulerian digraphs containing a vertex of large degree. Define the maximum degree of a digraph to be the maximum of all in-degrees and out-degrees of its vertices. We show that the diameter of an Eulerian digraph of order n and maximum degree Δ is at most n−Δ+3, and this bound is sharp. We also show that the bound can be improved for Eulerian digraphs with no 2-cycle to n−2Δ+O(Δ2/3), and we exhibit an infinite family of such digraphs of diameter n−2Δ+Ω(Δ1/2). We further show that for bipartite Eulerian digraphs with no 2-cycle the diameter is at most n−2Δ+3, and that this bound is sharp.

Complete characterization of almost Moore digraphs of degree three

It is well known that Moore digraphs do not exist except for trivial cases (degree 1 or diameter 1), but there are digraphs of diameter two and arbitrary degree which miss the Moore bound by one. No examples of such digraphs of diameter at least three are known, although several necessary conditions for their existence have been obtained. In this paper, we prove that digraphs of degree three and diameter k ≥ 3 which miss the Moore bound by one do not exist. © 2004 Wiley Periodicals, Inc. J Graph Theory 48: 112–126, 2005

Digraphs of degree two which miss the Moore bound by two

It is well known that Moore digraphs do not exist except for trivial cases (degree one or diameter one). Consequently, for a given maximum out-degree d and a given diameter, we wish to find a digraph whose order misses the Moore bound by the smallest possible ‘defect’. For diameter two and arbitrary degree there are digraphs which miss the Moore bound by one. No examples of such digraphs of diameter at least three are known, although several necessary conditions for their existence have been obtained. In the case of degree two, it has been shown that there are no digraphs of diameter greater than two and defect one. There are five nonisomorphic digraphs of degree two, diameter two and defect two. In this paper we prove that digraphs of degree two and diameter k⩾3 which miss the Moore bound by two do not exist.

Uniform homomorphisms of de Bruijn and Kautz networks

In this paper, we study the problem of homomorphisms of a general class of line digraphs. We show that the homomorphisms can always be defined using a partial binary operation on the alphabet whose letters form labels of the vertices. We apply these results to de Bruijn and Kautz (in short B/K) digraphs to characterize their uniform homomorphisms. For d non-prime, we describe algorithms for constructing non-trivial uniform homomorphisms of d-ary B/K digraphs of diameter D onto d-ary B/K digraphs of diameter D − 1. Using the properties of the uniform homomorphisms and shortest-path spanning trees of B/K digraphs, we also describe optimal emulations of Divide&Conquer computations on B/K digraphs.

Digraphs of degree 3 and order close to the moore bound

AbstractIt is known tht Moore digraphs of degree d > 1 and diameter k > 1 do not exist (see [20] or [5]). Furthermore, for degree 2, it is shown tht for l ≥ 3 there are no digraphs of order “close” to, i.e., one less than Moore bound [18]. In this paper, we shall consider digraphs of diameter k, degree 3 and number of vertices one less than Moore bound. We give a necessary condition for the existence of such digraphs and, using this condition, we deduce that such digraphs do not exist for infinitely many values of the diameter. © 1995 John Wiley & Sons, Inc.