Toroidal networks are generated as rotagraphs and some of their graph theoretical properties are described.

We give relatively simple proofs of both known and new results for the bandwidth of graphs involving direct products of paths and cycles. These include the rectangular lattice, the cylinder graph, the toroidal graph, and some more related results.

In this paper, we show that seven pages are sufficient for a book embedding of any toroidal graph.

The dimer model on a graph embedded in the torus can be interpreted as a collection of random self-avoiding loops. In this paper, we consider the uniform toroidal honeycomb dimer model. We prove that when the mesh of the graph tends to zero and the aspect of the torus is fixed, the winding number of the collection of loops converges in law to a two-dimensional discrete Gaussian distribution. Th...

In our recent paper W.S. Rossi, P. Frasca and F. Fagnani, “Average resistance of toroidal graphs”, SIAM Journal on Control and Optimization, 53(4):2541–2557, 2015, we studied how the average resistances of d-dimensional toroidal grids depend on the graph topology and on the dimension of the graph. Our results were based on the connection between resistance and Laplacian eigenvalues. In this not...

In many scientiic elds, three dimensional surfaces must be reconstructed from a given collection of its surface points. Applications for surface reconstruction exist in medical research and diagnosis as well as in design intensive disciplines. Fuchs, Kedem, and Uselton and Keppel show that surface reconstruction via triangulation can be reduced to the problem of nding a path in a toroidal graph...

A Schnyder wood is an orientation and coloring of the edges of a planar map satisfying a simple local property. We propose a generalization of Schnyder woods to graphs embedded on the torus with application to graph drawing. We prove several properties on this new object. Among all we prove that a graph embedded on the torus admits such a Schnyder wood if and only if it is an essentially 3conne...

It is well known that not all 3-connected planar graphs are hamiltonian. Whitney [10] proved that every triangulation of the sphere with no separating triangles is hamiltonian. Tutte [9] proved that every 4-connected planar graph has a Hamilton cycle. Extending Tutte's technique, Thomassen [8] proved that every 4-connected planar graph is in fact Hamilton connected. (A small omission in [8] was...