Odd crossing number vs ordinary crossings

A relation: 

• Which Crossing Number Is It Anyway? J. Pach and G. Tóth. Journal of Combinatorial Theory, Series B, Volume 80, Issue 2, November 2000, Pages 225-246.

How to get rid of even-times crossed edges: 

• Removing even crossings. Pelsmajer, Stefankovič, Schaefer. Journal of Combinatorial Theory Series B Volume 97 Issue 4, July, 2007.

• Michael J. Pelsmajer, Marcus Schaefer, Daniel Stefankovic: Removing Independently Even Crossings. SIAM J. Discrete Math. 24(2): 379-393 (2010)

And a BIG surprise after all: 

• Odd Crossing Number and Crossing Number Are Not the Same. Pelsmajer, Schaefer, Stefankovič. Discrete & Computational Geometry Volume 39 Issue 1, March 2008.

• Géza Tóth: Note on the Pair-crossing Number and the Odd-crossing Number. Discrete & Computational Geometry 39(4): 791-799 (2008)

The crossing number cr(G) of a graph G is the minimum possible number of edge-crossings in a drawing of G, the pair-crossing number pair-cr(G) is the minimum possible number of crossing pairs of edges in a drawing of G, and the odd-crossing number odd-cr(G) is the minimum number of pairs of edges that cross an odd number of times. Clearly, odd-cr(G) £ pair-cr(G) £ codd-cr(G)<=pair-cr(G)<=cr(G) . We construct graphs with 0855* pair-cr(G)>= odd-cr(G) . This improves the bound of Pelsmajer, Schaefer and Štefankovič. Our construction also answers an old question of Tutte. Slightly improving the bound of Valtr, we also show that if the pair-crossing number of G is k, then its crossing number is at most O(k ²/log ² k). 

Sylvester's Four point problem

• The Rectilinear Crossing Number of a Complete Graph and Sylvester's "Four Point Problem" of Geometric Probability (1994), by Edward R. Scheinerman, Herbert Wilf.