DMTCS Proceedings, 2005 European Conference on Combinatorics, Graph Theory and Applications (EuroComb '05)

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DMTCS Conference vol AE (2005), pp. 105-110


2005 European Conference on Combinatorics, Graph Theory and Applications (EuroComb '05)

Stefan Felsner (ed.)

DMTCS Conference Volume AE (2005), pp. 105-110

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
. 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. 105-110
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