Some sufficient conditions for the existence of kernels in infinite digraphs

A kernel N of a digraph D is an independent set of vertices of D such that for every w∈V(D)−N there exists an arc from w to N. If every induced subdigraph of D has a kernel, D is said to be a kernel perfect digraph. D is called a critical kernel imperfect digraph when D has no kernel but every proper induced subdigraph of D has a kernel. If F is a set of arcs of D, a semikernel modulo F of D is an independent set of vertices S of D such that for every z∈V(D)−S for which there exists an (S,z)-arc of D−F, there also exists an (z,S)-arc in D. In this work we show sufficient conditions for an infinite digraph to be a kernel perfect digraph, in terms of semikernel modulo F. As a consequence it is proved that symmetric infinite digraphs and bipartite infinite digraphs are kernel perfect digraphs. Also we give sufficient conditions for the following classes of infinite digraphs to be kernel perfect digraphs: transitive digraphs, quasi-transitive digraphs, right (or left)-pretransitive digraphs, the union of two right (or left)-pretransitive digraphs, the union of a right-pretransitive digraph with a left-pretransitive digraph, the union of two transitive digraphs, locally semicomplete digraphs and outward locally finite digraphs.     

On large vertex-symmetric digraphs

There is special interest in the design of large vertex-symmetric graphs and digraphs as models of interconnection networks for implementing parallelism. In these systems, a large number of nodes are connected with relatively few links and short paths between the nodes, and each node may execute the same communication software without modifications.In this paper, a method for obtaining new general families of large vertex-symmetric digraphs is put forward. To be more precise, from a k-reachable vertex-symmetric digraph and another (k+1)-reachable digraph related to the previous one, and using a new special composition of digraphs, new families of vertex-symmetric digraphs with small diameter are presented. With these families we obtain new vertex-symmetric digraphs that improve various values of the table of the largest known vertex-symmetric (Δ,D)-digraphs. The paper also contains the (Δ,D)-table for vertex-symmetric digraphs, for Δ≤13 and D≤12.     

On the restricted arc-connectivity of s-geodetic digraphs

For a strongly connected digraph D the minimum ,cardinality of an arc-cut over all arc-cuts restricted arc-connectivity λ′（D） is defined as the S satisfying that D - S has a non-trivial strong component D1 such that D - V（D1） contains an arc. Let S be a subset of vertices of D. We denote by w＋（S） the set of arcs uv with u ∈ S and v S, and by w-（S） the set of arcs uv with u S and v ∈ S. A digraph D = （V, A） is said to be λ′-optimal if λ′（D） =ξ′（D）, where ξ′（D） is the minimum arc-degree of D defined as ξ（D） = min {ξ′（xy） ： xy ∈ A}, and ξ′（xy） = min（｜ω＋（{x,y}）｜, ｜w-（{x,y}）｜, ｜w＋（x） ∪ w- （y） ｜, ｜w- （x） ∪ω＋ （y）｜}. In this paper a sufficient condition for a s-geodetic strongly connected digraph D to be λ′-optimal is given in terms of its diameter. Furthermore we see that the h-iterated line digraph Lh（D） of a s-geodetic digraph is λ′-optimal for certain iteration h.     

On the structure of strong 3-quasi-transitive digraphs

In this paper, D=(V(D),A(D)) denotes a loopless directed graph (digraph) with at most one arc from u to v for every pair of vertices u and v of V(D). Given a digraph D, we say that D is 3-quasi-transitive if, whenever u→v→w→z in D, then u and z are adjacent or u=z. In Bang-Jensen (2004) [3], Bang-Jensen introduced 3-quasi-transitive digraphs and claimed that the only strong 3-quasi-transitive digraphs are the strong semicomplete digraphs and strong semicomplete bipartite digraphs. In this paper, we exhibit a family of strong 3-quasi-transitive digraphs distinct from strong semicomplete digraphs and strong semicomplete bipartite digraphs and provide a complete characterization of strong 3-quasi-transitive digraphs.     

Characterization of eccentric digraphs

The eccentric digraphED(G) of a digraph G represents the binary relation, defined on the vertex set of G, of being ‘eccentric’; that is, there is an arc from u to v in ED(G) if and only if v is at maximum distance from u in G. A digraph G is said to be eccentric if there exists a digraph H such that G=ED(H). This paper is devoted to the study of the following two questions: what digraphs are eccentric and when the relation of being eccentric is symmetric.We present a characterization of eccentric digraphs, which in the undirected case says that a graph G is eccentric iff its complement graph is either self-centered of radius two or it is the union of complete graphs. As a consequence, we obtain that all trees except those with diameter 3 are eccentric digraphs. We also determine when ED(G) is symmetric in the cases when G is a graph or a digraph that is not strongly connected.     

Double-super-connected digraphs

A strongly connected digraph D is said to be super-connected if every minimum vertex-cut is the out-neighbor or in-neighbor set of a vertex. A strongly connected digraph D is said to be double-super-connected if every minimum vertex-cut is both the out-neighbor set of a vertex and the in-neighbor set of a vertex. In this paper, we characterize the double-super-connected line digraphs, Cartesian product and lexicographic product of two digraphs. Furthermore, we study double-super-connected Abelian Cayley digraphs and illustrate that there exist double-super-connected digraphs for any given order and minimum degree.     

Cayley Digraphs of Finite Simple Groups with Small Out-Valency

Abstract

For a group G and a subset S of G which does not contain the identity of G, the Cayley digraph Cay(G, S) is called normal if R(G) is normal in Aut(Γ). In this paper, we investigate the normality of Cayley digraphs of finite simple groups with out-valency 2 and 3. We give several sufficient conditions for such Cayley digraphs to be normal. By using this result, we consider the digraphical regular representations of finite simple groups.     

On the number of noncritical vertices in strongly connected digraphs

By definition, a vertex w of a strongly connected (or, simply, strong) digraph D is noncritical if the subgraph D — w is also strongly connected. We prove that if the minimal out (or in) degree k of D is at least 2, then there are at least k noncritical vertices in D. In contrast to the case of undirected graphs, this bound cannot be sharpened, for a given k, even for digraphs of large order. Moreover, we show that if the valency of any vertex of a strong digraph of order n is at least 3/4n, then it contains at least two noncritical vertices. The proof makes use of the results of the theory of maximal proper strong subgraphs established by Mader and developed by the present author. We also construct a counterpart of this theory for biconnected (undirected) graphs.     

A Family of Nonnormal Cayley Digraphs

We call a Cayley digraph Γ = Cay(G, S) normal for G if G _R , the right regular representation of G, is a normal subgroup of the full automorphism group Aut(Γ) of Γ. In this paper we determine the normality of Cayley digraphs of valency 2 on nonabelian groups of order 2p ² (p odd prime). As a result, a family of nonnormal Cayley digraphs is found. Received February 23, 1998, Revised September 25, 1998, Accepted October 27, 1998