Applications of the crossing number

We show that any graph of <italic>n</italic> vertices that can be drawn in the plane with no <italic>k</italic>+1 pairwise crossing edges has at most <italic>c<subscrpt>k</subscrpt>n</italic>log<supscrpt>2<italic>k</italic>−2</supscrpt><italic>n</italic> edges. This gives a partial answer to a dual version of a well-known problem of Avital-Hanani, Erdo&huml;s, Kupitz, Perles, and others. We also construct two point sets {<italic>p</italic><subscrpt>1</subscrpt>,…,<italic>p<subscrpt>n</subscrpt></italic>}, {<italic>q</italic><subscrpt>1</subscrpt>,…,<italic>q<subscrpt>n</subscrpt></italic>} in the plane such that any piecewise linear one-to-one mapping <italic>f</italic>:R<supscrpt>2</supscrpt>→R<supscrpt>2</supscrpt> with <italic>f(p<subscrpt>i</subscrpt>)=q<subscrpt>i</subscrpt></italic> (1≤<italic>i</italic>≤<italic>n</italic>) is composed of at least &OHgr;(<italic>n</italic><supscrpt>2</supscrpt>) linear pieces. It follows from a recent result of Souvaine and Wenger that this bound is asymptotically tight. Both proofs are based on a relation between the crossing number and the bisection width of a graph.