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arXiv · 2409.01723

Holes in Convex and Simple Drawings

Helena Bergold·Joachim Orthaber·Manfred Scheucher·Felix Schröder
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Abstract

Gons and holes in point sets have been extensively studied in the literature. For simple drawings of the complete graph a generalization of the Erd\H{o}s--Szekeres theorem is known and empty triangles have been investigated. We introduce a notion of $k$-holes for simple drawings and survey generalizations thereof, like empty $k$-cycles. We present a family of simple drawings without $4$-holes and prove a generalization of Gerken's empty hexagon theorem for convex drawings. A crucial intermediate step is the structural investigation of pseudolinear subdrawings in convex drawings. With respect to empty $k$-cycles, we show the existence of empty $4$-cycles in every simple drawing of $K_n$ and give a construction that admits only $\Theta(n^2)$ of them.

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Helena Bergold, Joachim Orthaber, Manfred Scheucher, Felix Schröder. 2024-09-03. Holes in Convex and Simple Drawings. https://doi.org/10.7155/jgaa.v29i3.2999

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