Removing Even Crossings on Surfaces

We give a new, topological proof that the weak Hanani-Tutte theorem is true on orientable surfaces and extend the result to nonorientable surfaces. That is, we show that if a graph G cannot be embedded on a surface S, then any drawing of G on S must contain two edges that cross an odd number of times. We apply the result and proof techniques to obtain new and old results about generalized thrackles, including that every bipartite generalized thrackle in a surface S can be embedded in 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 (an edge is even if it crosses every other edge an even number of times). From this result 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 show that ocr S ( G ) = cr S ( G ) whenever ocr S ( G ) ≤ 2 , for any surface S.