Norbert Seifter

Dominating Cartesian products of cycles

Abstract Let γ(G) be the domination number of a graph G and let G □ H denote the Cartesian product of graphs G and H. We prove that γ(X) = (Π m k = 1 n k ) (2m + 1) , where X = C1□C2□ … □ Cm and all n k = ¦C k ¦, 1 ⩽ k ⩽ m , are multiples of 2m + 1. The methods we use to prove this result immediately lead to an algorithm for finding minimum dominating sets of the considered graphs. Furthermore the domination numbers of products of two cycles are determined exactly if one factor is equal to C3, C4 or C5, respectively.

Constructing Infinite One-regular Graphs

A graph is said to be one-regular if its automorphism group acts regularly on the set of its arcs. A construction of an infinite family of infinite one-regular graphs of valency 4 is given. These graphs are Cayley graphs of almost abelian groups and hence of polynomial growth.

A survey on graphs with polynomial growth

In this paper we give an overview on connected locally finite transitive graphs with polynomial growth. We present results concerning the following topics: •Automorphism groups of graphs with polynomial growth.•Groups and graphs with linear growth.•S-transitivity.•Covering graphs.•Automorphism groups as topological groups.

A note on bounded automorphisms of infinite graphs

LetX be a connected locally finite graph with vertex-transitive automorphism group. IfX has polynomial growth then the set of all bounded automorphisms of finite order is a locally finite, periodic normal subgroup ofAUT(X) and the action ofAUT(X) onV(X) is imprimitive ifX is not finite. IfX has infinitely many ends, the group of bounded automorphisms itself is locally finite and periodic.

Automorphism Groups of Covering Graphs

For a large class of finite Cayley graphs we construct covering graphs whose automorphism groups coincide with the groups of lifted automorphisms. As an application we present new examples of 1/2-transitive and 1-regular graphs.

A note on the growth of transitive graphs

Abstract Let X be a locally finite, connected, infinite, transitive graph. We show that X has linear growth if and only if X is a strip.

Properties of graphs with polynomial growth

Abstract Let X be a connected locally finite transitive graph with polynomial growth. We prove that groups with intermediate growth cannot act transitively on X . Furthermore, it follows from this result that the automorphism group AUT( X ) is uncountable if and only if it contains a finitely generated subgroup with exponential growth which acts transitively on X . If X has valency at least three, we prove that X cannot be 8-transitive.

Highly arc transitive digraphs: reachability, topological groups

Let D be a locally finite, connected, 1-arc transitive digraph. It is shown that the reachability relation is not universal in D provided that the stabilizer of an edge satisfies certain conditions which seem to be typical for highly arc transitive digraphs. As an implication, the reachability relation cannot be universal in highly arc transitive digraphs with prime in- or out-degree.Two different aspects of the connection between highly arc transitive digraphs and the theory of totally disconnected locally compact groups are also considered.

Automorphism Groups of Graphs with Quadratic Growth

Let?be a graph with almost transitive group Aut(?) and quadratic growth. We show that Aut(?) contains an almost transitive subgroup isomorphic to the free abelian group Z2.

Groups acting on graphs with polynomial growth

Abstract In the first part of this paper we consider nilpotent groups G acting with finitely many orbits on infinite connected locally finite graphs X thereby showing that all α ϵ G of infinite order are automorphisms of type 2 of X . In the second part we investigate the automorphism groups of connected locally finite transitive graphs X with polynomial growth thereby showing that AUT( X ) is countable if and only if it is finitely generated and nilpotent-by-finite. In this case we also prove that X is contractible to a Cayley graph C ( G , H ) of a nilpotent group G (for some finite generating set H ) which has the same growth degree as X . If X is a transitive strip we show that AUT( X ) is uncountable if and only if it contains a finitely generated metabelian subgroup with exponential growth.