Title: Removing Even Crossings
Published: September 2005
Authors: Michael J. Pelsmajer, Marcus Schaefer, Daniel Štefankovic
Abstract: An edge in a drawing of a graph is called {\em 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 such that its even edges are not involved in any intersections. We give a new, and significantly simpler, proof of a slightly stronger statement. We show 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 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.
Full Paper: [pdf]