The connectivity of large digraphs and graphs

This paper studies the relation between the connectivity and other parameters of a digraph (or graph), namely its order n, minimum degree δ, maximum degree Δ, diameter D, and a new parameter l_pi;, 0 ≤ π ≤ δ ? 2, related with the number of short paths (in the case of graphs l₀ = ?(g ? 1)/2? where g stands for the girth). For instance, let G = (V,A) be a digraph on n vertices with maximum degree Δ and diameter D, so that n ≤ n(Δ, D) = 1 + Δ + Δ ² + … + Δ^D (Moore bound). As the main results it is shown that, if κ and λ denote respectively the connectivity and arc-connectivity of G, . Analogous results hold for graphs. © 1993 John Wiley & Sons, Inc.     

On the connectivity of randomm-orientable graphs and digraphs

We consider graphs and digraphs obtained by randomly generating a prescribed number of arcs incident at each vertex. We analyse their almost certain connectivity and apply these results to the expected value of random minimum length spanning trees and arborescences. We also examine the relationship between our results and certain results of Erdős and Rényi.     

Size of graphs and digraphs with given diameter and connectivity constraints

Acta Mathematica Hungarica - We determine the maximum size of a graph of given order, diameter, and edge-connectivity $$lambda$$ for $$2leq lambda leq 7$$ . This completes the determination of...     

Cayley digraphs and graphs

We shortly recall some definitions that involve refinements of connectivity, and a theorem of Y.O. Hamidoune. We consider some aspects of Cayley digraphs and vertex- and arc-transitive digraphs that he investigated.     

Degree and local connectivity in digraphs

It is shown that there is a digraphD of minimum outdegree 12m and μ(x, y; D)=11m, but every digraphD of minimum outdegreen contains verticesx ≠y withλ(x, y; D)≧n−1, whereμ(x, y; D) andλ(x, y; D) denote the maximum number of openly disjoint and edge-disjoint paths, respectively.     

Distance and size in digraphs

We determine the maximum number of arcs in an Eulerian digraph of given order and diameter. Our bound generalises a classical result on the maximum number of edges of an undirected graph of given order and diameter by Ore (1968) and Homenko and Ostroverhii? (1970). We further determine the maximum size of a bipartite digraph of given order and radius.     

Cycle extendability in graphs and digraphs

In 1990, Hendry conjectured that all chordal Hamiltonian graphs are cycle extendable, that is, the vertices of each non-Hamiltonian cycle are contained in a cycle of length one greater. Let A be a symmetric (0,1)-matrix with zero main diagonal such that A is the adjacency matrix of a chordal Hamiltonian graph. Hendry’s conjecture in this case is that every k×k principle submatrix of A that dominates a full cycle permutation k×k matrix is a principle submatrix of a (k+1)×(k+1) principle submatrix of A that dominates a (k+1)×(k+1) full cycle permutation matrix. This article generalizes the concept of cycle-extendability to S-extendable; that is, with S⊆{1,2,…,n} and G a graph on n vertices, G is S-extendable if the vertices of every non-Hamiltonian cycle are contained in a cycle length i greater, where i∈S. We investigate this concept in directed graphs and in particular tournaments, i.e., anti-symmetric matrices with zero main diagonal.     

Local-edge-connectivity in digraphs and oriented graphs

A digraph without any cycle of length two is called an oriented graph. The local-edge-connectivityλ(u,v) of two vertices u and v in a digraph or graph D is the maximum number of edge-disjoint u-v paths in D, and the edge-connectivity of D is defined as . Clearly, λ(u,v)?min{d⁺(u),d^-(v)} for all pairs u and v of vertices in D. Let δ(D) be the minimum degree of D. We call a graph or digraph D maximally edge-connected when λ(D)=δ(D) and maximally local-edge-connected when

λ(u,v)=min{d⁺(u),d^-(v)}     

Derangement action digraphs and graphs

A derangement of a set $X$ is a fixed-point-free permutation of $X$. Derangement action digraphs are closely related to group action digraphs introduced by Annexstein, Baumslag and Rosenberg in 1990. For a non-empty set $X$ and a non-empty subset $S$ of derangements of $X$, the derangement action digraph $\overrightarrow{DA}(X,S)$ has vertex set $X$, and an arc from $x$ to $y$ if and only if $y$ is the image of $x$ under the action of some element of $S$, so by definition it is a simple digraph. In common with Cayley graphs and Cayley digraphs, derangement action digraphs may be useful to model networks since the same routing and communication schemes can be implemented at each vertex. We prove that the family of derangement action digraphs contains all Cayley digraphs, all finite vertex-transitive simple graphs, and all finite regular simple graphs of even valency. We determine necessary and sufficient conditions on $S$ under which $\overrightarrow{DA}(X,S)$ may be viewed as a simple undirected graph of valency $\left|S\right|$. We investigate structural and symmetry properties of these digraphs and graphs, pose several open problems, and give many examples.     

Arbitrarily traceable graphs and digraphs

The work in this paper extends and generalizes earlier work by Ore on arbitrarily traceable Euler graphs, by Harary on arbitrarily traceable digraphs, by Chartrand and White on randomly n-traversable graphs, and by Chartrand and Lick on randomly Eulerian digraphs. Arbitrarily traceable graphs of mixed type are defined and characterized in terms of a class of forbidden graphs. Arbitrarily traceable digraphs of mixed type are also defined and a simply applied characterization is given for them.     

Ferrers digraphs and threshold graphs

We deduce a set of known characterizations of threshold graphs (Theorem 3) from a set of characterizations of Ferrers digraphs (Theorem 1) by investigating the connection between symmetric Ferrers digraphs and threshold graphs. A direct proof of Theorem 3 is easier than the one provided in here, but the purpose of this paper is to view Theorem 1 as an extension of Theorem 3 to the directed case (this extension point of view still holds on an algorithmic ground).     

Eigenvalues of graphs and digraphs

We show that given a digraph X there is a vertex-transitive digraph Y such that the minimal polynomial of X divides that of Y. Two consequences of this are that there exist vertex-transitive digraphs with nondiagonalizable adjacency matrices and that any algebraic integer is the eigenvalue of some vertex-transitive digraph.     

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.     

Strict refinement for graphs and digraphs

We show that any two cartesian factorizations of a connected graph have a strict common refinement, improving on the unique factorization theorem of G. Sabidussi. (The cartesian product is the product wherein two vertices are adjacent when they are adjacent in one coordinate and equal in all other coordinates.) Among the applications, we can deduce the strict refinement theorem for chain-finite posets, and (using a cartesian factorization algorithm of P. Winkler) we give a polynomial-time algorithm for cardinal factorization of connected finite posets.     

Isomorphism of circulant graphs and digraphs

Let S? {1, …, n?1} satisfy ?S = S mod n. The circulant graph G(n, S) with vertex set {v₀, v₁,…, v_n?1} and edge set E satisfies v_iv_j?E if and only if j ? i ∈ S, where all arithmetic is done mod n. The circulant digraph G(n, S) is defined similarly without the restriction S = ? S. Ádám conjectured that $\text{G(n, S) ? G(n, S′)}$ if and only if S = uS′ for some unit u mod n. In this paper we prove the conjecture true if n = pq where p and q are distinct primes. We also show that it is not generally true when n = p², and determine exact conditions on S that it be true in this case. We then show as a simple consequence that the conjecture is false in most cases when n is divisible by p² where p is an odd prime, or n is divisible by 2⁴.     

Fundamental groupoids of digraphs and graphs

We introduce the notion of fundamental groupoid of a digraph and prove its basic properties. In particular, we obtain a product theorem and an analogue of the Van Kampen theorem. Considering the category of (undirected) graphs as the full subcategory of digraphs, we transfer the results to the category of graphs. As a corollary we obtain the corresponding results for the fundamental groups of digraphs and graphs. We give an application to graph coloring.