Upper and lower bounds on long dual paths in line arrangements

Abstract

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/ log n). Our results also hold for pseudoline arrangements.