Planar crossing numbers of graphs of bounded genus. (English summary) Discrete Comput. Geom. 48 (2012), no. 2, 393–415.1432-0444 Summary: “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.
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