18 research outputs found
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 productGâK2, can be described in terms of regular embeddings ofG. This allows us to âliftâ 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 âderivedâ maps by employing those of the âbaseâ maps. We apply these results to determining all orientable regular embeddings of the tensor productsKnâK2(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 ofKnâK2exist only ifnis a prime powerpl, and there are 2Ï(nâ1) orÏ(nâ1) isomorphism classes of such maps (whereÏis 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 HĂK, 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 mâ„3 and nâ„2 has become an attractive topic in this area. In this paper, we deal with the orientable regular embeddings of Km[n] for mâ„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 pâ„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 3âČ-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
New methods for finding minimum genus embeddings of graphs on orientable and non-orientable surfaces
The question of how to find the smallest genus of all embeddings of a given finite connected
graph on an orientable (or non-orientable) surface has a long and interesting history.
In this paper we introduce four new approaches to help answer this question, in both the
orientable and non-orientable cases. One approach involves taking orbits of subgroups of
the automorphism group on cycles of particular lengths in the graph as candidates for subsets
of the faces of an embedding. Another uses properties of an auxiliary graph defined
in terms of compatibility of these cycles. We also present two methods that make use
of integer linear programming, to help determine bounds for the minimum genus, and to
find minimum genus embeddings. This work was motivated by the problem of finding the
minimum genus of the Hoffman-Singleton graph, and succeeded not only in solving that
problem but also in answering several other open questions