A slightly better bound on the crossing number in terms of the pair-crossing number

Abstract

The crossing number of a graph $G$, ${\mbox{cr}}(G)$, is the minimum number of crossings, the pair-crossing number, ${\mbox{pcr}}(G)$, is the minimum number of pairs of crossing edges over all drawings of $G$. In this note we show that ${\mbox{cr}}(G)=O({\mbox{pcr}}(G)^{3/2}\log{\mbox{pcr}}(G))$, which is an improvement of the result of Matou\v{s}ek, by a log factor.