Note on the pair-crossing number and the odd-crossing number
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)  = ODD-CR(G). This improves the bound of Pels-majer, Schaefer and Stefankovi. c. 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).
crossing number drawings of a graph