Cell-paths in mono- and bichromatic line arrangements in the plane

Abstract

We show that in every arrangement of n red and blue lines-in general position and not all of the same color-there is a path through a linear number of cells where red and blue lines are crossed alternatingly (and no cell is revisited). When all lines have the same color, and hence the preceding alternating constraint is dropped, we prove that the dual graph of the arrangement always contains a path of length θ(n2).