Automorphisms and regular embeddings of merged Johnson graphs

Doklady Mathematics, 2008

Acta Mathematica Sinica-english Series, 2005

A graph is called a semi–regular graph if its automorphism group action on its ordered pair of adjacent vertices is semi–regular. In this paper, a necessary and sufficient condition for an automorphism of the graph Γ to be an automorphism of a map with the underlying graph Γ is obtained. Using this result, all orientation–preserving automorphisms of maps on surfaces (orientable and non–orientable) or just orientable surfaces with a given underlying semi–regular graph Γ are determined. Formulas for the numbers of non–equivalent embeddings of this kind of graphs on surfaces (orientable, non–orientable or both) are established, and especially, the non–equivalent embeddings of circulant graphs of a prime order on orientable, non–orientable and general surfaces are enumerated.

Journal of the London Mathematical Society, 1999

It is proved that a graph K has an embedding as a regular map on some closed surface if and only if its automorphism group contains a subgroup G which acts transitively on the oriented edges of K such that the stabiliser G e of every edge e is dihedral of order 4 and the stabiliser G v of each vertex is a dihedral group the cyclic subgroup of index 2 of which acts regularly on the edges incident with. Such a regular embedding can be realised on an orientable surface if and only if the group G has a subgroup H of index 2 such that H v is the cyclic subgroup of index 2 in G v. An analogous result is proved for orientably-regular embeddings.

Discrete Mathematics, 2000

A simple combinatorial construction is given which takes as its imput a regular graph of valency k such that the convex closure of two points at distance two is the complete bipartite graph K3;3 and whose output is a regular graph of valency 2k + 1. It is shown that the sequence of graphs obtained by starting with the graph with one point and no edges and applying the construction recursively is the family of bipartite dual polar space of type DSO + (2n; 2).

European Journal of Combinatorics, 2007

A 2-cell embedding of a graph in an orientable closed surface is called regular if its automorphism group acts regularly on arcs of the embedded graph. The aim of this and of the associated consecutive paper is to give a classification of regular embeddings of complete bipartite graphs K n,n , where n = 2 e . The method involves groups G which factorise as a product XY of two cyclic groups of order n so that the two cyclic factors are transposed by an involutory automorphism. In particular, we give a classification of such groups G. Employing the classification we investigate automorphisms of these groups, resulting in a classification of regular embeddings of K n,n based on that for G. We prove that given n = 2 e , e ≥ 3 there are, up to map isomorphism, exactly 2 e−2 +4 regular embeddings of K n,n . Our analysis splits naturally into two cases depending on whether the group G is metacyclic or not.

Doklady Mathematics, 2011

Discrete Mathematics, 2000

A simple combinatorial construction is given which takes as its imput a regular graph of valency k such that the convex closure of two points at distance two is the complete bipartite graph K3;3 and whose output is a regular graph of valency 2k + 1. It is shown that the sequence of graphs obtained by starting with the graph with one point and no edges and applying the construction recursively is the family of bipartite dual polar space of type DSO + (2n; 2).

Doklady Mathematics, 2008

We consider nonoriented graphs without loops and multiple edges. For a vertex a of a graph Γ , by Γ i (a) we denote the i-neighborhood of a , i.e., the subgraph induced by Γ on the set of all vertices at distance i from a. We set [ a ] = Γ 1 (a) and a ⊥ = { a } ∪ [ a ]. For a graph Γ and a , b ∈ Γ , we denote the number of vertices in [ a ] ∩ [ b ] by µ (a , b) (by λ (a , b)) if a and b are distance 2 apart (adjacent) in Γ. If Γ is a graph of diameter d and I ⊂ {1, 2, …, d } , then Γ I denotes the graph on the same vertices as Γ in which two vertices u and w are adjacent if and only if d (u , w) ∈ I .

European Journal of Combinatorics, 2010

The aim of this paper is to complete a classification of regular orientable embeddings of complete bipartite graphs K n,n , where n = 2 e. The method involves groups G which factorise as a product G = XY of two cyclic groups of order n such that the two cyclic factors are transposed by an involutory automorphism. In particular, we give a classification of such groups G in the case where G is not metacyclic. We prove that for each n = 2 e , e ≥ 3, there are up to map isomorphism exactly four regular embeddings of K n,n such that the automorphism group G preserving the surface orientation and the bi-partition of vertices is a non-metacyclic group, and that there is one such embedding when n = 4.

Journal of Algebraic Combinatorics, 2000

In , the classification problem of regular embeddings of a given graph was described in terms of pure group theory. With the philosophy in [5], we shall classify the regular embeddings of simple graphs of order pq for any two primes p and q (not necessarily distinct) in this paper (see Theorem 4.8). The classification is based on the direct analysis of the structure of the arc-regular subgroups with the cyclic stabilizers of the automorphism groups of such graphs. Our analysis is independent of the classification of primitive permutation groups of degree p or degree pq. It is also independent of the classification of the arc-transitive graphs of order pq (p = q).

Journal of Algebraic Combinatorics, 2016

Extending earlier results of Godsil and of Dobson and Malnič on Johnson graphs, we characterise those merged Johnson graphs J = J(n, k) I which are Cayley graphs, that is, which are connected and have a group of automorphisms acting regularly on the vertices. We also characterise the merged Johnson graphs which are not Cayley graphs but which have a transitive group of automorphisms with vertex-stabilisers of order 2. Even though these merged Johnson graphs are all vertex-transitive, we show that only relatively few of them are Cayley graphs or have a transitive group of automorphisms with vertex-stabilisers of order 2.

European Journal of Combinatorics, 2005

In this paper, we classify all regular embeddings of the complete multipartite graphs K p,..., p for a prime p into orientable surfaces. Also, the same work is done for the regular embeddings of the lexicographical product of any connected arc-transitive graph of prime order q with the complement of the complete graph of prime order p, where q and p are not necessarily distinct. Lots of regular maps found in this paper are Cayley maps.

2017

In 1975, Babai characterized which abstract groups can be realized as the automorphism groups of planar graphs. In this paper, we give a more detailed and understandable description of these groups. We describe stabilizers of vertices in connected planar graphs as the class of groups closed under the direct product and semidirect products with symmetric, dihedral and cyclic groups. The automorphism group of a connected planar graph is then obtained as a semidirect product of a direct product of these stabilizers with a spherical group. The formulation of the main result is new and original. Moreover, it gives a deeper in the structure of the groups. As a consequence, automorphism groups of several subclasses of planar graphs, including 2-connected planar, outerplanar, and series-parallel graphs, are characterized. Our approach translates into a quadratic-time algorithm for computing the automorphism group of a planar graph which is the first such algorithm described in detail.

Filomat, 2013

A bicirculant is a graph admitting an automorphism whose cyclic decomposition consists of two cycles of equal length. In this paper we consider automorphisms of the so-called Tahacjn graphs, a family of pentavalent bicirculants which are obtained from the generalized Petersen graphs by adding two additional perfect matchings between the two orbits of the above mentioned automorphism. As a corollary, we determine which Tabacjn graphs are vertex-transitive.

Ars Mathematica Contemporanea

Regular embeddings of cycles with multiple edges have been reappearing in the literature for quite some time, both in and outside topological graph theory. The present paper aims to draw a complete picture of these maps by providing a detailed description, classification, and enumeration of regular embeddings of cycles with multiple edges on both orientable and non-orientable surfaces. Most of the results have been known in one form or another, but here they are presented from a unique viewpoint based on finite group theory. Our approach brings additional information about both the maps and their automorphism groups, and also gives extra insight into their relationships.