On a theorem of Halin (with Wilfried Imrich)

Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg, 2016

This note presents a new, elementary proof of a generalization of a theorem of Halin to graphs with unbounded degrees, which is then applied to show that every connected, countably infinite graph G, with ℵ0 ≤ | Aut(G)| < 2 ℵ 0 and subdegree-finite automorphism group, has a finite set F of vertices that is setwise stabilized only by the identity automorphism. A bound on the size of such sets, which are called distinguishing, is also provided. To put this theorem of Halin and its generalization into perspective, we also discuss several related non-elementary, independent results and their methods of proof.

Graphs and Combinatorics, 1989

Let X be a connected locally finite graph with vertex-transitive automorphism group. If X has polynomial growth then the set of all bounded automorphisms of finite order is a locally finite, periodic normal subgroup of AUT(X) and the action of AUT(X) on V(X) is imprimitive if X is not finite. If X has infinitely many ends, the group of bounded automorphisms itself is locally finite and periodic.

2016

A group of permutations G of a set V is k-distinguishable if there exists a partition of V into k parts such that only the identity permutation in G fixes setwise all of the cells of the partition. The least cardinal number k such that (G, V) is kdistinguishable is its distinguishing number D(G, V). In particular, a graph Γ is kdistinguishable if its automorphism group Aut(Γ) satisfies D(Aut(Γ), V Γ) ≤ k. Various results in the literature demonstrate that when an infinite graph fails to have some property, then often some finite subgraph is similarly deficient. In this paper we show that whenever an infinite connected graph Γ is not k-distinguishable (for a given cardinal k), then it contains a ball B of finite radius whose distinguishing number is at least k. Moreover, this lower bound cannot be sharpened, since for any integer k ≥ 3 there exists an infinite, locally finite, connected graph Γ that is not k-distinguishable but in which every ball of finite radius is k-distinguishable. In the second half of this paper we show that a large distinguishing number for an imprimitive graph Γ is traceable to a high distinguishing number either of a block of imprimitivity or of the induced action of Aut(Γ) on the corresponding system of imprimitivity. The distinguishing numbers of infinite primitive graphs have been examined in detail in a previous paper by the authors together with T. W. Tucker.

2011

The distinguishing number of a group G acting faithfully on a set V is the least number of colors needed to color the elements of V so that no non-identity element of the group preserves the coloring. The distinguishing number of a graph is the distinguishing number of its full automorphism group acting on its vertex set. A connected graph Γ is said to have connectivity 1 if there exists a vertex α∈ VΓ such that Γ∖{α} is not connected. For α∈ V, an orbit of the point stabilizer G_α is called a suborbit of G. We prove that every connected primitive graph with infinite diameter and countably many vertices has distinguishing number 2. Consequently, any infinite, connected, primitive, locally finite graph is 2-distinguishable; so, too, is any infinite primitive group with finite suborbits. We also show that all denumerable vertex-transitive graphs of connectivity 1 and all Cartesian products of connected denumerable graphs of infinite diameter have distinguishing number 2. All of our re...

arXiv: Combinatorics, 2013

A group of permutations G of a set V is k-distinguishable if there exists a partition of V into k parts such that only the identity permutation in G fixes setwise all of the cells of the partition. The least cardinal number k such that (G,V) is k-distinguishable is its distinguishing number. In particular, a graph X is k-distinguishable if its automorphism group Aut(X) has distinguishing number at most k in its action on the vertices of X. Various results in the literature demonstrate that when an infinite graph fails to have some property, then often some finite subgraph is similarly deficient. In this paper we show that whenever an infinite connected graph X is not k-distinguishable (for a given cardinal k), then it contains a ball B of finite radius whose distinguishing number is at least k. Moreover, this lower bound cannot be sharpened, since for any integer k greater than 3 there exists an infinite, locally finite, connected graph X that is not k-distinguishable but in which e...

The Electronic Journal of Combinatorics

A group of permutations $G$ of a set $V$ is $k$-distinguishable if there exists a partition of $V$ into $k$ cells such that only the identity permutation in $G$ fixes setwise all of the cells of the partition. The least cardinal number $k$ such that $(G,V)$ is $k$-distinguishable is its distinguishing number $D(G,V)$. In particular, a graph $\Gamma$ is $k$-distinguishable if its automorphism group $\rm{Aut}(\Gamma)$ satisfies $D(\rm{Aut}(\Gamma),V\Gamma)\leq k$.Various results in the literature demonstrate that when an infinite graph fails to have some property, then often some finite subgraph is similarly deficient. In this paper we show that whenever an infinite connected graph $\Gamma$ is not $k$-distinguishable (for a given cardinal $k$), then it contains a ball of finite radius whose distinguishing number is at least $k$. Moreover, this lower bound cannot be sharpened, since for any integer $k \geq 3$ there exists an infinite, locally finite, connected graph $\Gamma$ that is no...

Discrete Mathematics, 2013

The (countable) perturbed existentially closed graph S (Gordinowicz, 2010 [5]) was introduced by the second author as a solution to a problem stated by Bonato (Problem 20 in Cameron [3]). The graph S is not isomorphic to the Rado graph, nevertheless it has the NN c property in the sense that subgraphs induced by the neighbourhood and by the non-neighbourhood of each vertex of S are isomorphic to S. The graph S is given explicitly and is also uniquely -up to an isomorphism -characterized by a perturbed existential closure property (Gordinowicz, 2010 [5]). In the paper we characterize isomorphisms of finite, induced subgraphs of S which can be extended to global automorphisms.

An automorphism on a graph G is a bijective mapping on the vertex set V (G), which preserves the relation of adjacency between any two vertices of G. An automorphism g fixes a vertex v if g maps v onto itself. The stabilizer of a set S of vertices is the set of all automorphisms that fix vertices of S. A set F is called fixing set of G, if its stabilizer is trivial. The fixing number of a graph is the cardinality of a smallest fixing set. The fixed number of a graph G is the minimum k, such that every k-set of vertices of G is a fixing set of G. A graph G is called a k-fixed graph if its fixing number and fixed number are both k. In this paper, we study the fixed number of a graph and give construction of a graph of higher fixed number from graph with lower fixed number. We find bound on k in terms of diameter d of a distance-transitive k-fixed graph.

Proceedings of the American Mathematical Society, 1989

Proceedings of the American Mathematical Society, 2012

We prove a 1978 1978 conjecture of Richard Weiss in the case of groups with composition factors of bounded rank. Namely, we prove that there exists a function g : N × N → N g: \mathbb {N} \times \mathbb {N} \to \mathbb {N} such that, for Γ \Gamma a connected G G -vertex-transitive, G G -locally primitive graph of valency at most d d , if G G has no alternating groups of degree greater than r r as sections, then a vertex stabiliser in G G has size at most g ( r , d ) g(r,d) .