Planar bipartite graphs can be represented as touching graphs of horizontal and vertical segments in $\mathbb{R}^2$. We study a generalization in space: touching graphs of axis-aligned rectangles in $\mathbb{R}^3$, and prove that planar 3-colorable graphs can be represented this way. The result implies a characterization of corner polytopes previously obtained by Eppstein and Mumford. A by-product of our proof is a distributive lattice structure on the set of orthogonal surfaces with given skeleton. Further, we study representations by axis-aligned non-coplanar rectangles in $\mathbb{R}^3$ such that all regions are boxes. We show that the resulting graphs correspond to octahedrations of an octahedron. This generalizes the correspondence between planar quadrangulations and families of horizontal and vertical segments in $\mathbb{R}^2$ with the property that all regions are rectangles.

翻译：平面二分图可表示为$\mathbb{R}^2$中水平线段与垂直线段的接触图。我们考虑其空间推广：$\mathbb{R}^3$中轴对齐矩形的接触图，并证明所有平面三色可着色图均可如此表示。该结果蕴含了Eppstein与Mumford此前得到的角多面体刻画。证明的一个副产品是在给定骨架的正交曲面集上建立了分配格结构。此外，我们研究$\mathbb{R}^3$中所有区域均为盒状且由非共面轴对齐矩形构成的表示，并证明所得图对应八面体的八面体剖分。这推广了平面四边形剖分与$\mathbb{R}^2$中所有区域均为矩形的水平/垂直线段族之间的对应关系。