Consider a $D$-dimensional hypercubic lattice, i.e. a graph $H$ embedded in ${\mathbb R}^D$ where the vertices have integer coordinates $ (x_1,...,x_D) \in {\mathbb Z}^D$ and edges are between all pairs $ \{ (x_1,...,x_i,...,x_D), (x_1,...,x_i+1,...,x_D)\}$.
From this graph $H$, build a second graph $S$ the following way.
consider the family of vertices of $H$, embedded in the $D-1$-dimensional hyperplanes $S_t$ defined by $\sum_{i=1}^D x_i = t$ with $t \in {\mathbb Z}$
consider that there is an edge between two vertices in $S_t$ if, in $H$, those two vertices have a common neighbour in $S_{t+1}$ or/and $S_{t-1}$.
Obviously $S = \cup_t S_t$ is a disconnected graph with each $S_t$ being a $D-1$ dimensional layer.
I remarked that for $D = 2, 3$, and apparently also for $D=4$, the $S_t$ are actually $D-1$-hypersphere packing contact graphs.
My question is, for what values of $D$ does this hold?
