Grid designs

TL;DR

This paper investigates conditions under which complete graphs can be decomposed into grid graphs formed by Cartesian products of paths or cycles, providing new existence results and constructions using finite field arithmetic.

Contribution

It establishes new existence results for grid graph decompositions, including specific cases like $C_n imes C_n$ and $P_4 imes P_4$, using finite field methods.

Findings

Toroidal grid-graphs $C_n imes C_n$ admit $G$-designs when $n$ is an odd prime or its square.

The grid $P_4 imes P_4$ admits a $G$-design, enabling applications to puzzle scrambling.

The grid $P_3 imes P_3$ does not admit a $G$-design.

Abstract

We define a grid graph $G$ as a Cartesian product of path-graphs $P_n$ or cycle-graphs $C_n$ as shown in Figure 1, and we ask, when can the edge set of a complete graph be expressed as a disjoint union of graphs isomorphic to $G$? That is, we are asking for which grid graphs a $G$-design exists, where a $G$-design is defined as 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.