Removing even crossings

11 years 11 months ago
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...
Michael J. Pelsmajer, Marcus Schaefer, Daniel Stef
Added 15 Dec 2010
Updated 15 Dec 2010
Type Journal
Year 2007
Where JCT
Authors Michael J. Pelsmajer, Marcus Schaefer, Daniel Stefankovic
Comments (0)