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arXiv · 1410.4126

The intersection graph of the disks with diameters the sides of a convex $n$-gon

Clemens Huemer·Pablo Pérez-Lantero
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Abstract

Given a convex polygon of $n$ sides, one can draw $n$ disks (called side disks) where each disk has a different side of the polygon as diameter and the midpoint of the side as its center. The intersection graph of such disks is the undirected graph with vertices the $n$ disks and two disks are adjacent if and only if they have a point in common. We prove that for every convex polygon this graph is planar. Particularly, for $n=5$, this shows that for any convex pentagon there are two disks among the five side disks that do not intersect, which means that $K_5$ is never the intersection graph of such five disks. For $n=6$, we then have that for any convex hexagon the intersection graph of the side disks does not contain $K_{3,3}$ as subgraph.

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Clemens Huemer, Pablo Pérez-Lantero. 2016-06-16. The intersection graph of the disks with diameters the sides of a convex $n$-gon. https://arxiv.org/abs/1410.4126

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