Removing even crossings

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 T´oth 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. 1 The Hanani-Tutte Theorem In 1970 Tutte published his paper “Toward a Theory of Crossing Numbers” [16] containing the following beautiful theorem. In any planar drawing of a non-plan...

• Type: Journal
• Year: 2007
• Where: JCT
• Authors: Michael J. Pelsmajer, Marcus Schaefer, Daniel Stefankovic
• Keywords: JCT 2007, Odd Crossing Number, Simpler Proof, Stronger Statement