On perfect coverings of two-dimensional grids

TL;DR

This paper investigates perfect multiple coverings of two-dimensional integer grids using algebraic methods, establishing conditions for their periodicity and exploring algorithmic aspects of their detection.

Contribution

It introduces an algebraic framework for analyzing perfect coverings, linking their structure to periodicity via formal power series and polynomial factors.

Findings

Perfect multiple coverings have non-trivial periodizers.

Configurations with certain polynomial factors are necessarily periodic.

Many grid coverings are proven to be periodic under specific conditions.

Abstract

We study perfect multiple coverings in translation invariant graphs with vertex set $\mathbb{Z}^2$ using an algebraic approach. In this approach we consider any such covering as a two-dimensional binary configuration which we then express as a two-variate formal power series. Using known results, we conclude that any perfect multiple covering has a non-trivial periodizer, that is, there exists a non-zero polynomial whose formal product with the power series presenting the covering is a two-periodic configuration. If a non-trivial periodizer has line polynomial factors in at most one direction, then the configuration is known to be periodic. Using this result, we find many setups where perfect multiple coverings of infinite grids are necessarily periodic. We also consider some algorithmic questions on finding perfect multiple coverings.