2005 European Conference on Combinatorics, Graph Theory and Applications (EuroComb '05)
Stefan Felsner (ed.)
DMTCS Conference Volume AE (2005), pp. 105110
author:  Michael J. Pelsmajer, Marcus Schaefer and Daniel Štefankovič 

title:  Removing Even Crossings 
keywords:  Hanani's theorem, Tutte's theorem, even crossings, crossing number, odd crossing number, independent odd crossing number 
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.

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reference:  Michael J. Pelsmajer and Marcus Schaefer and Daniel Štefankovič (2005), Removing Even Crossings, in 2005 European Conference on Combinatorics, Graph Theory and Applications (EuroComb '05), Stefan Felsner (ed.), Discrete Mathematics and Theoretical Computer Science Proceedings AE, pp. 105110 
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