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

Classification of regular embeddings of hypercubes of odd dimension

By a regular embedding of a graph into a closed surface we mean a 2-cell embedding with the automorphism group acting regularly on flags. Recently, Kwon and Nedela [Non-existence of nonorientable regular embeddings of n-dimensional cubes, Discrete Math., to appear] showed that no regular embeddings of the n-dimensional cubes Q, into nonorientable surfaces exist for any positive integer n > 2. In 1997, Nedela and Skoviera [Regular maps from voltage assignments and exponent groups, European J. Combin. 18 (1997) 807-823] presented a construction giving for each solution of the congrumce e(2) equivalent to 1 (mod n) a regular embedding M-e of the hypercube Q(n) into an orientable surface. It was conjectured that all regular embeddings of Q(n) into orientable surfaces can be constructed in this way. This paper gives a classification of regular embeddings of hypercubes Q(n) into orientable surfaces for n odd, proving affirmatively the conjecture of Nedela and koviera for every odd n. (c) 2006 Elsevier B.V. All rights reserved.X1113sciescopu

Classification of Regular Embeddings of Hypercubes of Odd Dimension

By a regular embedding of a graph into an orientable surface we mean a 2cell embedding with the automorphism group acting regularly on arcs. In 1997 Nedela and ˇ Skoviera [Europ. J. Comb. 18, 807-823] presented a construction giving for each solution of the congruence e 2 ≡ 1(mod n) a regular embedding Me of the hypercube Qn. It was conjectured that all regular embeddings of Qn can be constructed in this way. This paper gives a classification of regular embeddings of hypercubes Qn for n odd. The conjecture is confirmed affirmatively for every odd n.