Packing and coloring r-bounded axis-parallel rectangles

TL;DR

This paper presents a simplified proof establishing a linear bound on the piercing number of r-bounded axis-parallel rectangles in relation to their independence number, improving previous bounds and deriving a ratio bound for chromatic and clique numbers.

Contribution

It provides a simpler proof for a linear bound on the transversal number of r-bounded rectangles, refining earlier results and extending to ratio bounds between chromatic and clique numbers.

Findings

Established a new bound: τ ≤ 2(r+1)(ν−1) + 1

Improved previous bounds on piercing numbers for r-bounded rectangles

Derived a constant factor bound for χ/ω ratio in r-bounded families

Abstract

Let $\mathcal{R}$ be a family of axis-parallel rectangles in the plane. The transversal number $\tau(\mathcal{R})$ is the minimum number of points needed to pierce all the rectangles. The independence number $\nu(\mathcal{R})$ is the maximum number of pairwise disjoint rectangles. Given a positive real number $r$, we say that $\mathcal{R}$ is an r-bounded family if, for any rectangle in $\mathcal{R}$, the aspect ratio of the longer side over the shorter side is at most $r$. Gy\'arf\'as and Lehel asked if it is possible to bound the transversal number $\tau(\mathcal{R})$ with a linear function of the independence number $\nu(\mathcal{R})$. Ahlswede and Karapetyan claimed a positive answer for the particular case of $r$-bounded families, but without providing proof. Chudnovsky et al. confirmed the result proving the bound $\tau \leq (14 + 2r^2) \nu$. This note aims at giving a simple proof of $\tau \leq 2(r+1)(\nu-1) + 1$, slightly improving the previous results. As a consequence of this new approach, we also deduce a constant factor bound for the ratio $\frac{\chi}{\omega}$ in the case of $r$-bounded family.