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.

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