| Abstract: |
An edge in a drawing of a graph is called {\em even} if
it intersects every other edge of the graph an even number of times.
Pach and Toth 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 (the only proof we know of not to use 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. The paper
begins with a disarmingly simple proof of a weak (but standard) version
of the theorem by Hanani and Tutte. |