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

TL;DR

This paper improves the upper bound on the crossing number of a graph in terms of its pair-crossing number, adding a logarithmic factor to previous bounds, advancing understanding of graph crossing properties.

Contribution

It presents a tighter bound on the crossing number relative to the pair-crossing number, refining prior results by Matoušek with an additional logarithmic factor.

Findings

Established that cr(G)=O(pcr(G)^{3/2} log pcr(G))

Improved the theoretical bound on crossing number

Refined previous results by Matoušek

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.