We study the existence of plane substructures in drawings of the $d$-dimensional hypercube graph $Q_d$. We construct drawings of $Q_d$ which contain no plane subgraph with more than $2d-2$ edges, no plane path with more than $2d-3$ edges, and no plane matching of size more than $2d-4$. On the other hand, we prove that every rectilinear drawing of $Q_d$ with vertices in convex position contains a plane path of length $d$ (if $d$ is odd) or $d-1$ (if $d$ is even). We also prove that if a graph $G$ is a plane subgraph of every drawing of $Q_d$ for a sufficiently large $d$, then $G$ is necessarily a forest of caterpillars. Lastly, we give a short proof of a generalization of a result by Alpert et al. [Cong. Numerantium, 2009] on the maximum rectilinear crossing number of $Q_d$.

翻译：本文研究$d$维超立方体图$Q_d$的绘制中平面子结构的存在性问题。我们构造了$Q_d$的若干绘制，其中不包含边数超过$2d-2$的平面子图、边数超过$2d-3$的平面路径，以及规模超过$2d-4$的平面匹配。另一方面，我们证明每个顶点处于凸位置的$Q_d$直线绘制必然包含长度为$d$（当$d$为奇数时）或$d-1$（当$d$为偶数时）的平面路径。我们还证明：若图$G$是充分大维度$d$下$Q_d$所有绘制中均存在的平面子图，则$G$必然是毛虫树的森林。最后，我们对Alpert等人[Cong. Numerantium, 2009]关于$Q_d$最大直线交叉数结果的推广定理给出了简洁证明。