# Example to show pairwise crossing number is not equal to crossing number

A common point of two edges in a graph drawing that is not an incident vertex is called a crossing.

The crossing number $cr(G)$ is defined to be the minimum number of crossings in any drawing of $G$.

The pairwise crossing number of $G$, denoted with $pcr(G)$, is the minimum number of pairs of edges $(e_1, e_2) \in E \times E$, $e_1 \neq e_2$ such that $e_1$ and $e_2$ determine at least one crossing, over all drawings of $G$.

Can you give an example to show pairwise crossing number is not equal to crossing number? i.e. $pcr(G)\neq cr(G).$

• According to the definition, $pcr(G)\leq cr(G)$, however, it is not unknown whether $pcr(G)=cr(G)$. – Rupei Xu Apr 28 '15 at 15:12
• Surprisingly, I just noticed that In 1995, in the Open Problem session of the AMS Conference on Topological Graph Theory, Bojan Mohar posted the problem of whether $cr(G)=pcr(G)$ for all $G$. Kratochvıl and Matousek shows that in general, given a drawing, it need not be possible to eliminate multiple crossings of pairs without introducing new crossing pairs.Existing results are mainly on estimating the upper bound of $cr(G)$ using $pcr(G).$ But in general, it is still open. – Rupei Xu May 11 '15 at 7:56