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 $O(n^{5/3})$.

翻译：设$\mathbb{R}^3$中一个以凸多边形为面的多面体曲面~$\mathcal{C}$是图~$G$的侧面接触表示，若$G$的顶点与$\mathcal{C}$的面之间存在双射，使得相邻顶点对应的多边形恰好是在$\mathcal{C}$中共享完整公共边的多边形。我们证明$K_{3,8}$具有侧面接触表示，而$K_{3,250}$不具有。后一结论表明，具有侧面接触表示的$n$顶点图的边数受$O(n^{5/3})$约束。