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 os and Guy by showing that κ(n,e)n ² /e ³ tends to a positive constant as n→∈fty and n l e l 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 Simonovits. Received January 27, 1999, and in revised form March 23, 1999.