Gray-code path embedding of a grid in a hypercube

The order $00,01,11,10$ is a Gray code through the four vertices of $Q_2$. Applying this identification independently in $n$ coordinate pairs makes $[4{]}^n=P_4^n$ a spanning subgraph of the hypercube graph $Q_{2n}$. Therefore every vertex boundary in $Q_{2n}$ contains the corresponding boundary in $[4{]}^n$.