Determining whether there exists a graph such that its crossing number and pair crossing number are distinct is an important open problem in geometric graph theory. We show that \(\textit{cr}(G)=O(\mathop {\textrm{pcr}}(G)^{3/2})\) for every graph G, improving the previous best bound by a logarithmic factor. Answering a question of Pach and Tóth, we prove that the bisection width (and, in fact, the cutwidth as well) of a graph G with degree sequence \(d_1,d_2,\dots ,d_n\) satisfies \(\mathop {\textrm{bw}}(G)=O\big (\sqrt{\mathop {\textrm{pcr}}(G)+\sum _{k=1}^n d_k^2}\big )\) . Then we show that there is a constant \(C\ge 1\) such that the following holds: For any graph G of order n and any set S of at least \(n^C\) points in general position on the plane, G admits a straight-line drawing which maps the vertices to points of S and has no more than \(O\left( \log n\cdot \left( \mathop {\textrm{pcr}}(G)+\sum _{k=1}^n d_k^2\right) \right) \) crossings. Our proofs rely on a slightly modified version of a separator theorem for string graphs by Lee, which might be of independent interest.
