Removing even crossings on surfaces

In this paper we investigate how certain results related to the Hanani–Tutte theorem can be extended from the plane to surfaces. We give a simple topological proof that the weak Hanani–Tutte theorem is true on arbitrary surfaces, both orientable and non-orientable. We apply these results and the proof techniques to obtain new and old results about generalized thrackles, including that every bipartite generalized thrackle on a surface S can be embedded on S . We also extend to arbitrary surfaces a result of Pach and Tóth that allows the redrawing of a graph so as to remove all crossings with even edges. From this we can conclude that cr S ( G ) , the crossing number of a graph G on surface S , is bounded by 2 ocr S ( G ) 2 , where ocr S ( G ) is the odd crossing number of G on surface S . Finally, we prove that ocr S ( G ) = cr S ( G ) whenever ocr S ( G ) ≤ 2 , for any surface S .