Let G be a multigraph with n vertices and e>4n edges, drawn in the
plane such that any two parallel edges form a simple closed curve with at least
one vertex in its interior and at least one vertex in its exterior. Pach and
T\'oth (2018) extended the Crossing Lemma of Ajtai et al. (1982) and Leighton
(1983) by showing that if no two adjacent edges cross and every pair of
nonadjacent edges cross at most once, then the number of edge crossings in G
is at least αe3/n2, for a suitable constant α>0. The situation
turns out to be quite different if nonparallel edges are allowed to cross any
number of times. It is proved that in this case the number of crossings in G
is at least αe2.5/n1.5. The order of magnitude of this bound
cannot be improved.Comment: Appears in the Proceedings of the 26th International Symposium on
Graph Drawing and Network Visualization (GD 2018