A Counterexample to Wegner's Conjecture for Axis-Parallel Rectangles

2026-06-16

AI Summary

This study addresses Wegner’s conjecture concerning the relationship between the piercing number $\tau(\mathcal{R})$ and the size $\nu(\mathcal{R})$ of a maximum pairwise disjoint subfamily for families $\mathcal{R}$ of axis-aligned rectangles. By constructing a triangle-free rectangle intersection graph whose independence number is at most one quarter of its number of vertices, the authors present the first explicit counterexample to the conjecture. The constructed family satisfies $\tau(\mathcal{R}) \geq 2\nu(\mathcal{R})$, and further improves the known lower bound on the integrality gap of the standard point-piercing linear programming relaxation to $2.21$, establishing $\tau(\mathcal{R}) \geq 2.21\,\nu(\mathcal{R})$. This work combines techniques from combinatorial geometry and graph theory to achieve a breakthrough in understanding the interplay between independent sets and piercing numbers in rectangle intersection graphs.

Abstract

In 1965, Wegner conjectured that every finite family \(\mathcal R\) of axis-parallel rectangles in the plane satisfies \(τ(\mathcal R) \le 2ν(\mathcal R)-1\), where \(τ(\mathcal R)\) denotes the minimum number of points needed to pierce all rectangles in \(\mathcal R\), and \(ν(\mathcal R)\) denotes the maximum size of a pairwise disjoint subfamily. Over the last six decades, the conjecture has motivated a long line of work: it has been verified for several special classes of rectangle families, and the best known general upper bounds have been progressively improved, but the conjecture itself had remained open. We give an explicit counterexample. More precisely, we construct a triangle-free rectangle-intersection graph on \(n\) vertices whose independence number is at most \(n/4\). Since the graph is triangle-free, no point of the plane can lie in three rectangles; hence every piercing point hits at most two rectangles. Consequently, \(τ(\mathcal R) \ge n/2 \ge 2ν(\mathcal R)\), contradicting Wegner's conjectured bound. We also give a slightly more general construction for which \(τ(\mathcal R) \ge 2.21ν(\mathcal R)\). This shows that the standard point relaxation, equivalently the clique relaxation, for the Maximum Independent Set of Rectangles problem has integrality gap at least \(2.21\).

Problem

• Wegner's conjecture
• axis-parallel rectangles
• piercing number
• independence number
• rectangle-intersection graph

Innovation

• Wegner's conjecture
• axis-parallel rectangles
• piercing number
• independence number
• integrality gap