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

Computer Science > Computational Geometry [Submitted on 16 Jun 2026] Title:A Counterexample to Wegner's Conjecture for Axis-Parallel Rectangles View PDF HTML (experimental)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 \(2.21\). Current browse context: cs.CG References & Citations Loading... Bibliographic and Citation Tools Bibliographic Explorer (What is the Explorer?) Connected Papers (What is Connected Papers?) Litmaps (What is Litmaps?) scite Smart Citations (What are Smart Citations?) Code, Data and Media Associated with this Article alphaXiv (What is alphaXiv?) CatalyzeX Code Finder for Papers (What is CatalyzeX?) DagsHub (What is DagsHub?) Gotit.pub (What is GotitPub?) Hugging Face (What is Huggingface?) ScienceCast (What is ScienceCast?) Demos Recommenders and Search Tools Influence Flower (What are Influence Flowers?) CORE Recommender (What is CORE?) arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs.