Bounds for rectilinear crossing numbers
Journal of Graph Theory
Which crossing number is it anyway?
Journal of Combinatorial Theory Series B
JCDCG '98 Revised Papers from the Japanese Conference on Discrete and Computational Geometry
Pfaffian graphs, T-joins and crossing numbers
Combinatorica
Odd crossing number is not crossing number
GD'05 Proceedings of the 13th international conference on Graph Drawing
Removing even crossings on surfaces
European Journal of Combinatorics
Crossing number of graphs with rotation systems
GD'07 Proceedings of the 15th international conference on Graph drawing
Removing Independently Even Crossings
SIAM Journal on Discrete Mathematics
Connectivity augmentation in planar straight line graphs
European Journal of Combinatorics
Removing independently even crossings
GD'09 Proceedings of the 17th international conference on Graph Drawing
Hanani-Tutte and monotone drawings
WG'11 Proceedings of the 37th international conference on Graph-Theoretic Concepts in Computer Science
GD'11 Proceedings of the 19th international conference on Graph Drawing
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An edge in a drawing of a graph is called even if it intersects every other edge of the graph an even number of times. Pach and Toth proved that a graph can always be redrawn so that its even edges are not involved in any intersections. We give a new and significantly simpler proof of the stronger statement that the redrawing can be done in such a way that no new odd intersections are introduced. We include two applications of this strengthened result: an easy proof of a theorem of Hanani and Tutte (the only proof we know of not to use Kuratowski's theorem), and the new result that the odd crossing number of a graph equals the crossing number of the graph for values of at most 3. The paper begins with a disarmingly simple proof of a weak (but standard) version of the theorem by Hanani and Tutte.