Planar Crossing Numbers of Graphs of Bounded Genus

Pach and Tóth proved that any n -vertex graph of genus g and maximum degree d has a planar crossing number at most c g dn , for a constant c >1. We improve on this result by decreasing the bound to O ( dgn ), and also prove that our result is tight within a constant factor. Our proof is constructive and yields an algorithm with time complexity O ( dgn ). As a consequence of our main result, we show a relation between the planar crossing number and the surface crossing number.