Shaofei Du

Regular embeddings of complete multipartite graphs

In this paper, we classify all regular embeddings of the complete multipartite graphs Kp,...,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.

A Classification of Regular Embeddings of Graphs of Order a Product of Two Primes

In this paper, we classify the regular embeddings of arc-transitive simple graphs of order pq for any two primes p and q (not necessarily distinct) into orientable surfaces. Our classification is obtained by direct analysis of the structure of arc-regular subgroups (with cyclic vertex-stabilizers) of the automorphism groups of such graphs. This work is independent of the classification of primitive permutation groups of degree p or degree pq for p ≠ q and it is also independent of the classification of the arc-transitive graphs of order pq for p ≠ q.

2-Groups that factorise as products of cyclic groups, and regular embeddings of complete bipartite graphs

We classify those 2-groups G which factorise as a product of two disjoint cyclic subgroups A and B , transposed by an automorphism of order 2 . The case where G is metacyclic having been dealt with elsewhere, we show that for each e ≥ 3 there are exactly three such non-metacyclic groups G with ∣ A ∣ = ∣ B ∣ = 2 e , and for e = 2 there is one. These groups appear in a classification by Berkovich and Janko of 2 -groups with one non-metacyclic maximal subgroup; we enumerate these groups, give simpler presentations for them, and determine their automorphism groups.

2-Arc-Transitive regular covers of K_{n,n} - nK_2 with the covering transformation group Z_p^2

In 2014, Xu and Du classified all regular covers of a complete bipartite graph K n ,  n minus a matching, denoted by K n ,  n  −  n K 2 , whose covering transformation group is cyclic and whose fibre-preserving automorphism group acts 2-arc-transitively. In this paper, a further classification is achieved for all the regular covers of K n ,  n  −  n K 2 , whose covering transformation group is isomorphic to Z p 2 with p a prime and whose fibre-preserving automorphism group acts 2 -arc-transitively. Actually, there are only few covers with these properties and it is shown that all of them are covers of K 4, 4  − 4 K 2 .

Semisymmetric graphs of order 2p3

A simple undirected graph is said to be semisymmetric if it is regular and edge-transitive but not vertex-transitive. Every semisymmetric graph is a bipartite graph with two parts of equal size. It was proved in Folkman (1967) [14] that there exist no semisymmetric graphs of order 2p and 2p^2, where p is a prime. The classification of semisymmetric graphs of order 2pq was given in Du and Xu (2000) [12], for any distinct primes p and q. Our long term goal is to determine all the semisymmetric graphs of order 2p^3, for any prime p. All these graphs @C are divided into two subclasses: (I) Aut(@C) acts unfaithfully on at least one bipart; and (II) Aut(@C) acts faithfully on both biparts. This paper gives a group theoretical characterization for Subclass (I) and based on this characterization, we shall give a complete classification for this subclass in our further research.

A classification of primer hypermaps with a product of two primes number of hyperfaces

A hypermap is a cellular embedding of a connected bipartite graph G into a compact and connected surface S without border. The vertices of G lie in distinct partitions, colored black and white, and are called respectively the hypervertices and hyperedges of the hypermap, while the connected regions of GS are the hyperfaces. A hypermap H is called orientable if the underlying surface S is orientable. An orientable hypermap is called regular if its orientation preserving automorphism group G acts regularly on the flags (hypervertex, hyperedge and hyperface incident triples), and further, it is called (face-)primer if G induces a faithful action on their hyperfaces. In Breda dAzevedo and Fernandes (2011), a classification of the primer hypermap with a prime number of hyperfaces is given. In this paper, a classification will be given, for all primer hypermaps with a product of two primes hyperfaces, see Theorem5.1.

On the orientably-regular embeddings of graphs of order prime-cube

This paper characterizes the automorphism group G of the orientably-regular embeddings of simple graphs of order prime-cube p 3 . Our main result will be a starting point for classifying all such embeddings. Moreover, by using some known results, a partial classification is given, when G contains a Sylow p -subgroup of order p 5 .

Nonorientable regular embeddings of graphs of order p2

A map is called regular if its automorphism group acts regularly on the set of all flags (incident vertex-edge-face triples). An orientable map is called orientably regular if the group of all orientation-preserving automorphisms is regular on the set of all arcs (incident vertex-edge pairs). If an orientably regular map admits also orientation-reversing automorphisms, then it is regular, and is called reflexible. A regular embedding and orientably regular embedding of a graph G are, respectively, 2-cell embeddings of G as a regular map and orientably regular map on some closed surface. In Du et al. (2004) [7], the orientably regular embeddings of graphs of order pq for two primes p and q (p may be equal to q) have been classified, where all the reflexible maps can be easily read from the classification theorem. In [11], Du and Wang (2007) classified the nonorientable regular embeddings of these graphs for p q. In this paper, we shall classify the nonorientable regular embeddings of graphs of order p^2 where p is a prime so that a complete classification of regular embeddings of graphs of order pq for two primes p and q is obtained. All graphs in this paper are connected and simple.