  Given a line arrangement $\cal A$ with $n$ lines, we show that there exists a
path of length $n^2/3 - O(n)$ in the dual graph of $\cal A$ 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 $\Omega (n^2/ \log n)$. Our results also hold for pseudoline
arrangements.
