Partitioning the vertices of a torus into isomorphic subgraphs

Let $H$ be an induced subgraph of the torus $C_k^m$. We show that when $k \ge 3$ is even and $|V(H)|$ divides some power of $k$, then for sufficiently large $n$ the torus $C_k^n$ has a perfect vertex-packing with induced copies of $H$. On the other hand, disproving a conjecture of Gruslys, we show that when $k$ is odd and not a prime power, then there exists $H$ such that $|V(H)|$ divides some power of $k$, but there is no $n$ such that $C_k^n$ has a perfect vertex-packing with copies of $H$. We also disprove a conjecture of Gruslys, Leader and Tan by exhibiting a subgraph $H$ of the $k$-dimensional hypercube $Q_k$, such that there is no $n$ for which $Q_n$ has a perfect edge-packing with copies of $H$.

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