A better bound for the pair-crossing number

Abstract

The crossing number cr(G) of a graph G is the minimum possible number of edge crossings in a drawing of G, and the pair-crossing number pair-cr(G) is the minimum possible number of crossing pairs of edges in a drawing of G. Clearly, pair-cr(G) ≤ cr(G). We show that for any graph G, cr(G) = O(pair-cr(G)7/4 log 3/2 (pair-cr(G))).