Classification and Galois conjugacy of Hamming maps

We show that for each d>0 the d-dimensional Hamming graph H(d,q) has an orientably regular surface embedding if and only if q is a prime power p^e. If q>2 there are up to isomorphism \phi(q-1)/e such maps, all constructed as Cayley maps for a d-dimensional vector space over the field of order q. We show that for each such pair d, q the corresponding Belyi pairs are conjugate under the action of the absolute Galois group, and we determine their minimal field of definition. We also classify the orientably regular embedding of merged Hamming graphs for q>3

Regular Embeddings of Canonical Double Coverings of Graphs

AbstractThis paper addresses the question of determining, for a given graphG, all regular maps havingGas their underlying graph, i.e., all embeddings ofGin closed surfaces exhibiting the highest possible symmetry. We show that ifGsatisfies certain natural conditions, then all orientable regular embeddings of its canonical double covering, isomorphic to the tensor productG⊗K2, can be described in terms of regular embeddings ofG. This allows us to “lift” the classification of regular embeddings of a given graph to a similar classification for its canonical double covering and to establish various properties of the “derived” maps by employing those of the “base” maps. We apply these results to determining all orientable regular embeddings of the tensor productsKn⊗K2(known as the cocktail-party graphs) and of then-dipolesDn, the graphs consisting of two vertices and n parallel edges joining them. In the first case we show, in particular, that regular embeddings ofKn⊗K2exist only ifnis a prime powerpl, and there are 2φ(n−1) orφ(n−1) isomorphism classes of such maps (whereφis Euler's function) according to whetherlis even or odd. Forleven an interesting new infinite family of regular maps is discovered. In the second case, orientable regular embeddings ofDnexist for each positive integern, and their number is a power of 2 depending on the decomposition ofninto primes

Regular maps with nilpotent automorphism groups

AbstractWe study regular maps with nilpotent automorphism groups in detail. We prove that every nilpotent regular map decomposes into a direct product of maps H×K, where Aut(H) is a 2-group and K is a map with a single vertex and an odd number of semiedges. Many important properties of nilpotent maps follow from this canonical decomposition, including restrictions on the valency, covalency, and the number of edges. We also show that, apart from two well-defined classes of maps on at most two vertices and their duals, every nilpotent regular map has both its valency and covalency divisible by 4. Finally, we give a complete classification of nilpotent regular maps of nilpotency class 2

On the orientable regular embeddings of complete multipartite graphs

AbstractLet Km[n] be the complete multipartite graph with m parts, while each part contains n vertices. The regular embeddings of complete graphs Km[1] have been determined by Biggs (1971) [1], James and Jones (1985) [12] and Wilson (1989) [23]. During the past twenty years, several papers such as Du et al. (2007, 2010) [6,7], Jones et al. (2007, 2008) [14,15], Kwak and Kwon (2005, 2008) [16,17] and Nedela et al. (2002) [20] contributed to the regular embeddings of complete bipartite graphs K2[n] and the final classification was given by Jones [13] in 2010. Since then, the classification for general cases m≥3 and n≥2 has become an attractive topic in this area. In this paper, we deal with the orientable regular embeddings of Km[n] for m≥3. We in fact give a reduction theorem for the general classification, namely, we show that if Km[n] has an orientable regular embedding M, then either m=p and n=pe for some prime p≥5 or m=3 and the normal subgroup Aut0+(M) of Aut+(M) preserving each part setwise is a direct product of a 3-subgroup Q and an abelian 3′-subgroup, where Q may be trivial. Moreover, we classify all the embeddings when m=3 and Aut0+(M) is abelian. We hope that our reduction theorem might be the first necessary approach leading to the general classification

Regular embeddings of complete bipartite maps: classification and enumeration

The regular embeddings of complete bipartite graphs Kn, n in orientable surfaces are classified and enumerated, and their automorphism groups and combinatorial properties are determined. The method depends on earlier classifications in the cases where n is a prime power, obtained in collaboration with Du, Kwak, Nedela and koviera, together with results of Itô, Hall, Huppert and Wielandt on factorisable groups and on finite solvable groups. <br/