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

TL;DR

This paper studies the intersection graphs of disks constructed from the sides of convex polygons, proving planarity for all polygons and specific subgraph exclusions for pentagons and hexagons.

Contribution

It establishes that the intersection graph of side disks of any convex polygon is planar and identifies forbidden subgraphs for pentagons and hexagons.

Findings

The intersection graph of side disks of any convex polygon is planar.

For convex pentagons, the intersection graph cannot contain a complete graph of five vertices.

For convex hexagons, the intersection graph does not contain a bipartite subgraph K_{3,3}.

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.