Tutte has shown that every 4-connected planar graph contains a Hamilton cycle. Grünbaum and Nash-Williams independently conjectured that the same is true for toroidal graphs. In this paper, we prove that every 4-connected toroidal graph contains a Hamilton path. Partially supported by NSF grant DMS-9970514 Partially supported by NSF grants DMS-9970527 and DMS-0245530 Partially supported by RGC ...

It is shown that if a graph of n vertices can be drawn on the torus without edge crossings and the maximum degree of its vertices is at most d, then its planar crossing number cannot exceed cdn, where c is a constant. This bound, conjectured by Brass, cannot be improved, apart from the value of the constant. We strengthen and generalize this result to the case when the graph has a crossing-free...

The chromatic polynomials of some families of quadrangulations of the torus can be found explicitly. The method, known as ‘bracelet theory’ is based on a decomposition in terms of representations of the symmetric group. The results are particularly appropriate for studying the limit curves of the chromatic roots of these families. In this paper these techniques are applied to a family of quadra...

We show that the cop number of toroidal graphs is at most 3. This resolves a conjecture by Schroeder from 2001 which implicit in question Andreae 1986.

Abstract. The average effective resistance of a graph is a relevant performance index in many applications, including distributed estimation and control of network systems. In this paper, we study how the average resistance depends on the graph topology and specifically on the dimension of the graph. We concentrate on d-dimensional toroidal grids and we exploit the connection between resistance...