In this note we study embeddings of Cayley graphs of right groups on surfaces. We characterize those right groups which have a toroidal but no planar Cayley graph, such that the generating system of the right group has a minimal generating system of the group as a factor.

The Manhattan Street Network (MSN) is a digraph resulting from labeling the edges of a torus with directions. We study some of the basic properties of the MSN labeling. We prove a lower bound on the stretch of any path due to such a labeling, and that the maximum stretch of the MSN meets this bound. We also prove that the diameter of the MSN is not optimal as there exists another labeling of th...

We show that every 4-representative graph embedding in the double torus contains a noncontractible cycle which separates the surface into two pieces. This improves a result of Zha and Zhao for general orientable surfaces, in which the same conclusion holds for 6-representative graph embeddings. Noncontractible separating cycles have been studied because they provide a way to do induction on the...

We consider the problem of gossiping in several important networks in as few rounds as possible. During a single round, each processor may send an unlimited size message to k neighbors, or receive messages from k neighbors, but a processor cannot both send and receive during the same round. The network architectures we consider are trees, cycles, grids, hypercubes, and toroidal (or " wrap-aroun...

Denote the n × n toroidal queen’s graph by Qn. We find its automorphism group Aut(Qn) for each positive integer n, showing that for n ≥ 6, Aut(Qn) is generated by the translations, the group of the square, the homotheties, and (for odd n) the automorphism (x, y) → (y + x, y − x). For each n we find the automorphism classes of edges of Qn, in particular showing that for n > 1, Qn is edge-transit...

A toroidal polyhex (resp. Klein-bottle polyhex) described by a string (p, q, t) arises from a p × q-parallelogram of a hexagonal lattice by a usual torus (resp. Klein bottle) boundary identification with a torsion t. A connected graph G admitting a perfect matching is kextendable if |V(G)| ≥ 2k + 2 and any k independent edges can be extended to a perfect matching of G. In this paper, we charact...

The interplay between structural chemistry, graph theory and knot theory is a rich area for chemists and mathematicians alike. The founder of topology, Johann Listing, noted the possibility of chiral knots in 1847; a year later Pasteur published his landmark paper on the optical activity of ammonium tartrate crystals. Indeed, knot theory began following early studies on molecular isomerism. Mor...

Schnyder woods are particularly elegant combinatorial structures with numerous applications concerning planar triangulations and more generally 3-connected planar maps. We propose a simple generalization of Schnyder woods from the plane to maps on orientable surfaces of any genus with a special emphasis on the toroidal case. We provide a natural partition of the set of Schnyder woods of a given...

Abstract. The energy of a simple graph G arising in chemical physics, denoted by E(G), is defined as the sum of the absolute values of eigenvalues of G. As the dimer problem and spanning trees problem in statistical physics, in this paper we propose the energy per vertex problem for lattice systems. In general for a type of lattices in statistical physics, to compute the entropy constant with t...

We classify the minimum volume smooth complex hyperbolic surfaces that admit smooth toroidal compactifications, and we explicitly construct their compactifications. There are five such surfaces and they are all arithmetic, i.e., they are associated with quotients of the ball by an arithmetic lattice. Moreover, the associated lattices are all commensurable. The first compactification, originally...