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arXiv · 2507.00313

Faces in rectilinear drawings of complete graphs

Martin Balko·Anna Brötzner·Fabian Klute·Josef Tkadlec
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Abstract

We initiate the study of extremal problems about faces in convex rectilinear drawings of~$K_n$, that is, drawings where vertices are represented by points in the plane in convex position and edges by line segments between the points representing the end-vertices. We show that if a convex rectilinear drawing of $K_n$ does not contain a common interior point of at least three edges, then there is always a face forming a convex 5-gon while there are such drawings without any face forming a convex $k$-gon with $k \geq 6$. A convex rectilinear drawing of $K_n$ is \emph{regular} if its vertices correspond to vertices of a regular convex $n$-gon. We characterize positive integers $n$ for which regular drawings of $K_n$ contain a face forming a convex 5-gon. To our knowledge, this type of problems has not been considered in the literature before and so we also pose several new natural open problems.

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Martin Balko, Anna Brötzner, Fabian Klute, Josef Tkadlec. 2025-06-30. Faces in rectilinear drawings of complete graphs. https://arxiv.org/abs/2507.00313

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