Crossing number, pair-crossing number, and expansion

The crossing number cr(G) of a graph G is the minimum possible number of edge crossings in a drawing of G in the plane, while the pair-crossing number pcr(G) is the smallest number of pairs of edges that cross in a drawing of G in the plane. While cr(G) ≥ pcr(G) holds trivially, it is not known whether a strict inequality can ever occur (this question was raised by Mohar and Pach and Toth). We aim at bounding cr(G) in terms of pcr(G). Using the methods of Leighton and Rao, Bhatt and Leighton, and Even, Guha and Schieber, we prove that cr(G) = O(log3 n(pcr(G) + ssqd(G))), where n = |V(G)| and ssqd(G) = Σ v∈V(G) deg G (v)2. One of the main steps is an analogy of the well-known lower bound cr(G) = Ω(b(G)2) - O(ssqd(G)), where b(G) is the bisection width of G, that is, the smallest number of edges that have to be removed so that no component of the resulting graph has more than 2/3 n vertices. We show that pcr(G) = Ω(b(G)2/log2 n) - O(ssqd(G)). We also prove by similar methods that a graph G with crossing number k = cr(G) > C√ssqd(G) m log2 n has a nonplanar subgraph on at most O(Δnm log2 n/k) vertices, where m is the number of edges, Δ is the maximum degree in G, and C is a suitable sufficiently large constant.