The intersection graph of the disks with diameters the sides of a convex $n$gon
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.
 Publication:

arXiv eprints
 Pub Date:
 October 2014
 DOI:
 10.48550/arXiv.1410.4126
 arXiv:
 arXiv:1410.4126
 Bibcode:
 2014arXiv1410.4126H
 Keywords:

 Mathematics  Metric Geometry;
 Computer Science  Computational Geometry