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 Erdos and Guy by showing that κ(n, e)n²/e³ tends to a positive constant as n → ∞ and n ≪ e ≪ n². 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 ce⁴/n³ (resp. ce⁵/n⁴), where c > 0 is a suitable constant. These results cannot be improved, apart from the value of the constant. This settles a question of Simonovits.