Every $k$-Quasi-Planar Graph Has Linearly Many Edges
How to prepare an accurate paper press kit? Illustrated guide
Ankan Sadhu · arXiv: 2611.01509v1 · Summary page
Why cover this story this week?
A crossing restriction forces sparse networks
Imagine drawing a network in the plane while forbidding a sufficiently large collection of edges from all crossing one another. Must the network have only linearly many edges as its vertex count grows? A new preprint answers yes for every fixed forbidden collection size. The drawings can be complicated: edges may cross repeatedly, and even edges sharing an endpoint may cross.
The proof changes the setting rather than simplifying each crossing. Removing a small disk around every vertex turns edges into arcs on a sphere with boundary circles. The authors then use a theorem about patterns on surfaces to bound the number of arcs, supported by a triangulation whose size grows linearly with the original network. The result shows how a global restriction on crossings can control network density, and how topology can supply a counting tool for combinatorial geometry.
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