A Crossing Lemma for the Pair-Crossing Number
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Abstract
The pair-crossing number of a graph G, pcr(G), is the minimum possible number of pairs of edges that cross each other (possibly several times) in a drawing of G. It is known that there is a constant c ≥ 1/64 such that for every (not too sparse) graph G with n vertices and m edges \({\mbox{pcr}}(G) \geq c \frac{m^3}{n^2}\). This bound is tight, up to the constant c. Here we show that c ≥ 1/34.2 if G is drawn without adjacent crossings.
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