Michael J. Pelsmajer ; Marcus Schaefer ; Daniel Štefankovič

Removing Even Crossings
dmtcs:3430 
Discrete Mathematics & Theoretical Computer Science,
January 1, 2005,
DMTCS Proceedings vol. AE, European Conference on Combinatorics, Graph Theory and Applications (EuroComb '05)

https://doi.org/10.46298/dmtcs.3430
Removing Even Crossings
Authors: Michael J. Pelsmajer ^{1}; Marcus Schaefer ^{2}; Daniel Štefankovič ^{3}
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Michael J. Pelsmajer;Marcus Schaefer;Daniel Štefankovič
1 Department of Applied Mathematics
2 Department of Computer Science [Chicago]
3 Department of Computer Science [Rochester]
An edge in a drawing of a graph is called $\textit{even}$ 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.
Pelsmajer, Michael J.; Schaefer, Marcus; Ĺ tefankoviÄ, Daniel, 2008, Odd Crossing Number And Crossing Number Are Not The Same, Discrete & Computational Geometry, 39, 13, pp. 442454, 10.1007/s004540089058x.