Upper and Lower Bounds on Long Dual-Paths in Line Arrangements

TL;DR

This paper establishes tight bounds on the maximum length of paths in the dual graph of line arrangements, including bounds for bicolored arrangements and the existence of long alternating paths.

Contribution

It provides the first tight bounds on the longest paths in the dual graphs of line arrangements and explores coloring strategies for long alternating paths.

Findings

Existence of a path of length approximately n^2/3 in the dual graph

Construction of arrangements with no long alternating paths

Existence of colorings with long alternating paths of length Omega(n^2 / log n)

Abstract

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.