Crossing Number of Toroidal Graphs

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 drawing on an orientable surface of higher genus and there is no restriction on the degrees of the vertices.