Upper and Lower Bounds on Long Dual Paths in Line Arrangements
Given a line arrangement \(\mathcal A\) with n lines, we show that there exists a path of length \(n^2/3 - O(n)\) in the dual graph of \(\mathcal A\) 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 \(\varOmega (n^2/ \log n)\). Our results also hold for pseudoline arrangements.
We thank Nieke Aerts, Stefan Felsner, Heuna Kim, and Piotr Micek for interesting and helpful discussions on the topic. We thank the anonymous reviewers for helpful comments.