Point-circle configurations and inscribability of polytopes and graphs
Presented by Dr. Gábor GéVAY
Type: Oral presentation
Track: Graph Theory
We call a graph $G$ inscribable of order $m$ (in brief, $m$-inscribable) if its vertices can be located on the sphere $S^2$ in such a way that for each vertex $v$, the vertices at distance $m$ from $v$ lie in a common plane ($m=1,2,...$). If, in particular, $G$ is planar and 3-connected, then, on account of Steinitz' characterization theorem, we can define the same property for a 3-polytope $P$ whose 1-skeleton is isomorphic to $G$. If the planes in the definition above are all distinct, then $G$ (respectively $P$) admits a point-circle configuration of type $(n_k)$, where $n$ is the number of vertices of $G$ and $k$ is the number of $m$th neighbours of $v$. Examples are the Platonic and Archimedean solids; they are all 1-inscribable, and some of them are 2-inscribable. In this talk we present, besides sporadic examples and finite classes, some infinite series of $m$-inscribable graphs and polyhedra. We also raise some open problems. The results presented here were obtained partly in joint work with Tomaz Pisanski.