Coloring problems on arrangements of pseudolines

TL;DR

The paper studies coloring problems in pseudoline arrangements, focusing on two crossing-coloring variants and pseudoline colorings. It proves that $n$ colors suffice to color crossings so that no color repeats on the boundary of any cell and also along any pseudoline, with the proofs framed via hypergraph colorings and acyclic orientations. It further investigates coloring pseudolines to avoid monochromatic crossings, showing NP-hardness for computing the pseudoline chromatic number $\text{chi}_{pl}$, and providing probabilistic bounds (via the Lovász Local Lemma) that achieve near-linear dependence on the arrangement size for avoiding high-degree monochromatic crossings. The work connects pseudoline coloring problems to the Erdős–Faber–Lovász conjecture and Turán-type results for ordinary points, offering both algorithmic implications and several open questions on tightening bounds and understanding dependencies on structural parameters like $mx(\mathcal{A})$. Overall, the results deepen the combinatorial understanding of colorings in pseudoline arrangements and suggest new directions for deterministic coloring procedures and complexity analysis.

Abstract

Arrangements of pseudolines are a widely studied generalization of line arrangements. They are defined as a finite family of infinite curves in the Euclidean plane, any two of which intersect at exactly one point. One can state various related coloring problems depending on the number $n$ of pseudolines. In this article, we show that $n$ colors are sufficient for coloring the crossings avoiding twice the same color on the boundary of any cell, or, alternatively, avoiding twice the same color along any pseudoline. We also study the problem of coloring the pseudolines avoiding monochromatic crossings.

Figures (12)

• Figure 1: A non-simple (a) and a simple (b) pseudoline arrangement together with a corresponding tiling (c).
• Figure 2: Coloring that fulfills the statements of Theorem \ref{['thm:crossing_coloring_face_respecting']} and Theorem \ref{['thm:crossing_coloring']} simultaneously.
• Figure 3: Proof idea of Lemma \ref{['lemma:pseudoline_face_incidence']}.
• Figure 4: The orientation induced by fixing a north cell is acyclic.
• Figure 5: (a): Example for conflict ancestors; (b), (c): Distinction between the cases of $F$ being an unbounded or a bounded cell in the proof of Lemma \ref{['lemma:bound_conflict_ancestors']}.

Theorems & Definitions (31)

• Theorem 1
• Theorem 2
• Theorem 3
• Lemma 1
• proof
• Lemma 2
• proof
• Lemma 3
• proof
• proof : Proof of Theorem \ref{['thm:crossing_coloring_face_respecting']}