A contact graph of a packing of closed balls is a graph with balls as vertices and pairs of tangent balls as edges. We prove that the average degree of the contact graph of a packing of balls (with possibly different radii) in ${\mathbb{R}}^3$ is not greater than 13.92. We also find new upper bounds for the average degree of contact graphs in ${\mathbb{R}}^4$ and ${\mathbb{R}}^5$.

中文翻译：

---

球填料的接触图

闭合球的堆积的接触图是以球为顶点，切线对为边缘的图。我们证明了球包装（半径可能不同）中接触图的平均程度${\mathbb{[R}}^3$不大于13.92。我们还找到了联系图平均程度的新上限${\mathbb{[R}}^4$ 和 ${\mathbb{[R}}^5$。