Side-Contact Representations with Convex Polygons in 3D: New Results for Complete Bipartite Graphs

TL;DR

This paper investigates side-contact representations of bipartite graphs with convex polygons in 3D, establishing bounds on which complete bipartite graphs can be represented and deriving an upper limit on the number of edges.

Contribution

It proves that certain complete bipartite graphs can or cannot be represented with convex polygon side-contacts in 3D, and provides an upper bound on the number of edges for such graphs.

Findings

$K_{3,8}$ has a side-contact representation.

$K_{3,250}$ does not have a side-contact representation.

The number of edges in such graphs is bounded by $O(n^{5/3})$.

Abstract

A polyhedral surface~$\mathcal{C}$ in $\mathbb{R}^3$ with convex polygons as faces is a side-contact representation of a graph~$G$ if there is a bijection between the vertices of $G$ and the faces of~$\mathcal{C}$ such that the polygons of adjacent vertices are exactly the polygons sharing an entire common side in~$\mathcal{C}$. We show that $K_{3,8}$ has a side-contact representation but $K_{3,250}$ has not. The latter result implies that the number of edges of a graph with side-contact representation and $n$ vertices is bounded by $O(n^{5/3})$.