New bounds on crossing numbers

The crossing number, cr(G), of a graph G is the least number of crossing points in any drawing of G in the plane. Denote by κ(n, e) the minimum of cr(G) taken over all graphs with n vertices and at least e edges. We prove a conjecture of P. Erdos and R. Guy by showing that κ(n, e)n ²/e ³ tends to a positive constant as n→∞ and n≪e≪n ². Similar results hold for graph drawings on any other surface of fixed genus. We prove better bounds for graphs satisfying some monotone properties. In particular, we show that if G is a graph with n vertices and e≥4n edges, which does not contain a cycle of length four (resp. six), then its crossing number is at least ce ⁴/n ³ (resp. ce ⁵/n ⁴), where c>0 is a suitable constant. These results cannot be improved, apart from the value of the constant. This settles a question of M. Simonovits.