Given a line arrangement A with n lines, we show that there exists a
path of length n2/3−O(n) in the dual graph of A formed by its
faces. This bound is tight up to lower order terms. For the bicolored version,
we describe an example of a line arrangement with 3k blue and 2k red lines
with no alternating path longer than 14k. Further, we show that any line
arrangement with n lines has a coloring such that it has an alternating path
of length Ω(n2/logn). Our results also hold for pseudoline
arrangements.Comment: 19 page