On the Pair-Crossing Number

By a drawing of a graph G, we mean a drawing in the plane such that vertices are represented by distinct points and edges by arcs. The crossing number cr(G) of a graph G is the minimum possible number of crossings in a drawing of G. The pair-crossing number pair-cr(G) of G is the minimum possible number of (unordered) crossing pairs in a drawing of G. Clearly, pair-cr(G) ≤ cr(G) holds for any graph G. Let f(k) be the maximum cr(G), taken over all graphs G with pair-cr(G) = k. Obviously, f(k) ≥ k. Pach and Tóth [2000] proved that f(k) ≤ 2k. Here we give a slightly better asymptotic upper bound f(k) = O(k/ log k). In case of x-monotone drawings (where each vertical line intersects any edge at most once) we get a better upper bound f(k) ≤ 4k4/3.