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

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

Enumerating reflexible 2-cell embeddings of connected graphs

Two 2-cell embeddings A +/-: X -> S and j: X -> S of a connected graph X into a closed orientable surface S are congruent if there are an orientation-preserving surface homeomorphism h on S and a graph automorphism gamma of X such that A +/- h = gamma j. A 2-cell embedding A +/- : X -> S of a graph X into a closed orientable surface S is described combinatorially by a pair (X; rho) called a map, where rho is a product of disjoint cycle permutations each of which is the permutation of the darts of X initiated at the same vertex following the orientation of S. The mirror image of a map (X; rho) is themap (X; rho (-1)), and one of the corresponding embeddings is called the mirror image of the other. A 2-cell embedding of X is reflexible if it is congruent to its mirror image. Mull et al. [Proc Amer Math Soc, 1988, 103: 321-330] developed an approach for enumerating the congruence classes of 2-cell embeddings of graphs into closed orientable surfaces. In this paper we introduce a method for enumerating the congruence classes of reflexible 2-cell embeddings of graphs into closed orientable surfaces, and apply it to the complete graphs, the bouquets of circles, the dipoles and the wheel graphs to count their congruence classes of reflexible or nonreflexible (called chiral) embeddings.111sciescopu

Congruence classes of orientable 2-cell embeddings of bouquets of circles and dipoles

Two 2-cell embeddings i : X -> S and j : X -> S of a connected graph X into a closed orientable surface S are congruent if there are an orientation-preserving surface homeomorphism h : S -> S and a graph automorphism gamma of X such that ih = gamma j. Mull et al. [Proc. Amer. Math. Soc. 103(1988) 321 330] developed an approach for enumerating the congruence classes of 2-cell embeddings of a simple graph (without loops and multiple edges) into closed orientable surfaces and as an application, two formulae of such enumeration were given for complete graphs and wheel graphs. The approach was further developed by Mull [J. Graph Theory 30(1999) 77-90] to obtain a formula for enumerating the congruence classes of 2-cell embeddings of complete bipartite graphs into closed orientable surfaces. By considering automorphisms of a graph as permutations on its dart set, in this paper Mull et al.'s approach is generalized to any graph with loops or multiple edges, and by using this method we enumerate the congruence classes of 2-cell embeddings of a bouquet of circles and a dipole into closed orientable surfaces.X111sciescopu