CrossingNumber is one of the most challenging algorithmic problems in topological graph theory, with applications to graph drawing and VLSI layout. No polynomial time approximation algorithm is known for this NP-Complete problem. We give in this paper a polynomial time approximation algorithm for the crossing number of toroidal graphs with bounded degree. In course of proving the algorithm we p...

In this paper, we build on the work of Alspach, Chen, and Dean [2] who showed that proving the hamiltonicity of the Cayley graph of the the dihedral group Dn reduces to showing that certain cubic, connected, bipartite graphs (called honeycomb toroidal graphs) are hamilton laceable; that is, any two vertices at odd distance from each other can be joined by a hamilton path. Alspach, Chen, and Dea...

We show that there is a linear-time algorithm to partition the edges of a planar graph into triangles. We show that the problem is also polynomial for toroidal graphs but NP-complete for k-planar graphs, where k > 8.

The vertex arboricity a(G) of a graph G is the minimum k such that V (G) can be partitioned into k sets where each set induces a forest. For a planar graph G, it is known that a(G) ≤ 3. In two recent papers, it was proved that planar graphs without k-cycles for some k ∈ {3, 4, 5, 6, 7} have vertex arboricity at most 2. For a toroidal graph G, it is known that a(G) ≤ 4. Let us consider the follo...

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...

We provide a description of unlabelled enumeration techniques, with complete proofs, for graphs that can be canonically obtained by substituting 2-pole networks for the edges of core graphs. Using structure theorems for toroidal and projectiveplanar graphs containing no K3,3-subdivisions, we apply these techniques to obtain their unlabelled enumeration.

We provide a description of unlabelled enumeration techniques, with complete proofs, for graphs that can be canonically obtained by substituting 2-pole networks for the edges of core graphs. Using structure theorems for toroidal and projectiveplanar graphs containing no K3,3-subdivisions, we apply these techniques to obtain their unlabelled enumeration.