Mark E. Watkins

The groups of the generalized Petersen graphs

1. Introduction . For integers n and k with 2 ≤ 2k n , the generalized Petersen graph G(n, k) has been defined in (8) to have vertex-set and edge-set E(G(n, k)) to consist of all edges of the form where i is an integer. All subscripts in this paper are to be read modulo n , where the particular value of n will be clear from the context. Thus G(n, k) is always a trivalent graph of order 2 n , and G (5, 2) is the well known Petersen graph. (The subclass of these graphs with n and k relatively prime was first considered by Coxeter ((2), p. 417ff.).)

Cayley maps

We present a theory of Cayley maps, i.e., embeddings of Cayley graphs into oriented surfaces having the same cyclic rotation of generators around each vertex. These maps have often been used to encode symmetric embeddings of graphs. We also present an algebraic theory of Cayley maps and we apply the theory to determine exactly which regular or edge-transitive tilings of the sphere or plane are Cayley maps or Cayley graphs. Our main goal, however, is to provide the general theory so as to make it easier for others to study Cayley maps.

On the action of non-Abelian groups on graphs

Abstract Finite groups may be classified as to whether or not they have regular representations as automorphism groups of graphs. Such a classification is begun here for non-Abelian groups (classification of Abelian groups being already in the literature) with a classification of the dihedral groups, the groups of order p 3 for p an odd prime (except for one group of order 27), and the generalized dicyclic groups. It is further shown that the class of groups (other than C 2 ) having the above regular representations is closed under the operation of direct product.

Graphical regular representations of non-abelian groups. I, II

In this paper, all groups and graphs considered are finite and all graphs are simple (in the sense of Tutte [8, p. 50]). If X is such a graph with vertex set V(X) and automorphism group A(X), we say that X is a graphical regular representation (GRR) of a given abstract group G if (I) G ^ ^ ( Z ) , a n d (II) A(X) acts on V(X) as a regular permutation group; that is, given u, v G V(X), there exists a unique <p £ A (X) for which <p(u) = v. That for any abstract group G there exists a graph X satisfying (I) is well-known (cf. [3]). The question of existence or non-existence of a GRR for a given abstract group G, however, has been settled to date only for relatively few classes of groups. This problem is the underlying motivation for this paper and its sequel. In Section 1 we introduce some essential notation and attempt a summary of what is known to date (with references to the literature) about the foregoing problem. In Section 2 some machinery involving at one time techniques from both group theory and graph theory is developed in order to facilitate proving, when true, that a given graph is indeed a GRR. These techniques are then used to prove the main result of the section:

Fragments and Automorphisms of Infinite Graphs

Es werden unendliche Graphen untersucht, die sich durch endliche trennende Mengen in mindestens zwei unendliche Teile zerlegen lassen. Dabei wird insbesondere der Frage nachgegangen, wie sich die Zusammenhangsstruktur und die Struktur der Automorphismengruppe wechselseitig beeinflussen.

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.

Infinite paths that contain only shortest paths

Let Γ be an infinite, locally finite, connected simple graph. A one-way (resp. two-way) infinite path of which every finite subpath is a shortest path is called a halfaxis (resp. an axis). The following are proved: (1) Given any vertex x and any end e of Γ, there exists a halfaxis belonging to e which originates at x (thus sharpening a classical resut of D. Konig, in “Theorie der endlichen und unendlichen graphen”, Akad. Verlagsgesellschaft, Leipzig, 1936); (2) every pair of distinct ends of Γ contains a pair of halfaxes whose union is an axis of Γ; (3) if H is a halfaxis and y is a vertex not on H, then some halfaxis originating at y meets H. If Γ is vertex-transitive, we prove: (4) every vertex lies on an axis, and (5) the complement of every axis has no finite component. We investigate vertex-transitive graphs having the following property: if A is an axis and y is a vertex not on A, then there exists an axis through y disjoint from A. Graphs with connectivity equal 1 and having this property are characterized.

On automorphism groups of Cayley graphs

LetXG,H denote the Cayley graph of a finite groupG with respect to a subsetH. It is well-known that its automorphism groupA(XG,H) must contain the regular subgroupLG corresponding to the set of left multiplications by elements ofG. This paper is concerned with minimizing the index [A(XG,H)∶LG] for givenG, in particular when this index is always greater than 1. IfG is abelian but not one of seven exceptional groups, then a Cayley graph ofG exists for which this index is at most 2. Nearly complete results for the generalized dicyclic groups are also obtained.

Graphical regular representations of alternating, symmetric, and miscellaneous small groups

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On the connectivities of finite and infinite graphs

Results byW. Mader and the authors on the connectivity of finite graphs are generalized to include infinite graphs. In the infinite case a distinction must be made concerning the distribution of finite and infinite components with respect to the separating sets. Results are obtained relating these components to the atoms.