What is the maximum number of intersections of the boundaries of a simple
m-gon and a simple n-gon, assuming general position? This is a basic
question in combinatorial geometry, and the answer is easy if at least one of
m and n is even: If both m and n are even, then every pair of sides may
cross and so the answer is mn. If exactly one polygon, say the n-gon, has
an odd number of sides, it can intersect each side of the m-gon at most n−1
times; hence there are at most mn−m intersections. It is not hard to
construct examples that meet these bounds. If both m and n are odd, the
best known construction has mn−(m+n)+3 intersections, and it is conjectured
that this is the maximum. However, the best known upper bound is only mn−(m+⌈6n⌉), for m≥n. We prove a new upper bound of
mn−(m+n)+C for some constant C, which is optimal apart from the value of
C.Comment: 18 pages, 24 figures. To appear in the proceedings of the 36th
International Symposium on Computational Geometry (SoCG 2020) in June 2016
(Eds. Sergio Cabello and Danny Chen