Computer Science > Computational Geometry

[Submitted on 16 Jun 2026]

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

Authors:Deepak Ajwani, Rishikesh Gajjala, Rajiv Raman, Saurabh Ray

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Abstract:In 1965, Wegner conjectured that every finite family (\mathcal R) of axis-parallel rectangles in the plane satisfies (\tau(\mathcal R) \le 2\nu(\mathcal R)-1), where (\tau(\mathcal R)) denotes the minimum number of points needed to pierce all rectangles in (\mathcal R), and (\nu(\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, (\tau(\mathcal R) \ge n/2 \ge 2\nu(\mathcal R)), contradicting Wegner's conjectured bound. We also give a slightly more general construction for which (\tau(\mathcal R) \ge 2.21\nu(\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).

https://doi.org/10.48550/arXiv.2606.17854

arXiv-issued DOI via DataCite

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Subjects:	Computational Geometry (cs.CG); Discrete Mathematics (cs.DM); Data Structures and Algorithms (cs.DS); Combinatorics (math.CO)
Cite as:	arXiv:2606.17854 [cs.CG]
	(or arXiv:2606.17854v1 [cs.CG] for this version)

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From: Rishikesh Gajjala [view email] [v1] Tue, 16 Jun 2026 12:26:17 UTC (29 KB)

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