Grid designs
Alon Danai, Joshua Kou, Andy Latto, Haran Mouli, James Propp
TL;DR
The paper studies grid designs, i.e., decompositions of complete graphs into edge-disjoint copies of grid graphs $G$ that are either torus grids $C_n \square C_n$ or ordinary grids $P_n \square P_n$. It proves explicit decompositions for torus grids when the side length is tied to odd primes, showing that $K_{p^2}$ can be partitioned into $(p^2-1)/4$ copies of $C_p \square C_p$ and $K_{p^4}$ into $(p^4-1)/4$ copies of $C_{p^2} \square C_{p^2}$ using finite-field constructions; for ordinary grids, it demonstrates that $K_9$ cannot be partitioned into three copies of $P_3 \square P_3$ but that $K_{16}$ can be partitioned into five copies of $P_4 \square P_4$ via a $\mathbb{F}_2[x]/(x^4+x+1)$-based grid mapping. The approach leverages algebraic methods on finite fields to produce explicit decompositions, connecting a combinatorial design problem to algebraic constructions and motivating a Connections puzzle interpretation. The paper also outlines open questions for other moduli and higher-dimensional grid products.
Abstract
We define a grid graph as a Cartesian product of path-graphs $P_n$ or cycle-graphs $C_n$, and define a grid design as a $G$-design where the graph $G$ is a grid graph, that is, a decomposition of a complete graph into edge-disjoint subgraphs isomorphic to $G$. We show that when $n$ is an odd prime or the square of an odd prime, the toroidal grid-graph $G = C_n \square C_n$ admits a $G$-design. In the less symmetrical case of products of path-graphs, we prove that $G = P_3 \square P_3$ does not admit a $G$-design but that $G = P_4 \square P_4$ does. This last result is the special case that motivated the present paper: a $P_4 \square P_4$-design corresponds to a way of successively scrambling a Connections puzzle so that each pair of words occurs adjacently exactly once. Our constructions use the arithmetic of finite fields.