Partitioning the vertices of a torus into isomorphic subgraphs

TL;DR

This paper investigates conditions under which large torus graphs can be perfectly partitioned into isomorphic induced subgraphs, revealing new cases where such packings are possible or impossible, and disproving existing conjectures.

Contribution

It establishes new results on perfect vertex-packings in tori and hypercubes, disproving conjectures by Gruslys and others regarding necessary conditions.

Findings

Perfect packings exist for even k when |V(H)| divides a power of k.

Counterexamples show packings do not always exist for odd non-prime power k.

Disproof of conjectures about perfect packings in hypercubes.

Abstract

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 $H$.