### Removing Even Crossings

*Michael J. Pelsmajer, Marcus Schaefer, Daniel Štefankovič*

#### Abstract

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 Tóth 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 (not using 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. We begin with a disarmingly simple proof of a weak (but standard) version of the theorem by Hanani and Tutte.Full Text: GZIP Compressed PostScript PostScript PDF original HTML abstract page