Lower bounds for identifying codes in some infinite grids

TL;DR

This paper establishes new lower bounds on the density of $r$-identifying codes in infinite square and hexagonal grids for small radius values, advancing understanding of code minimality in these structures.

Contribution

It provides novel lower bounds for the densities of identifying codes in infinite grids, specifically for small radii, which were previously unknown.

Findings

New lower bounds for code densities in square grids

New lower bounds for code densities in hexagonal grids

Improved understanding of minimal code sizes in infinite grids

Abstract

An $r$-identifying code on a graph $G$ is a set $C\subset V(G)$ such that for every vertex in $V(G)$, the intersection of the radius-$r$ closed neighborhood with $C$ is nonempty and unique. On a finite graph, the density of a code is $|C|/|V(G)|$, which naturally extends to a definition of density in certain infinite graphs which are locally finite. We present new lower bounds for densities of codes for some small values of $r$ in both the square and hexagonal grids.