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
NULL##NULL##NULL
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, 1-3, pp. 442-454, 10.1007/s00454-008-9058-x.