Optimal lower bound for 2-identifying code in the hexagonal grid

TL;DR

This paper proves that the known 2-identifying code in the hexagonal grid with density 4/19 is optimal, establishing a lower bound that cannot be improved further.

Contribution

It establishes the first proof that the 2-identifying code density of 4/19 in the hexagonal grid is the lowest possible, confirming its optimality.

Findings

The 2-identifying code density of 4/19 is proven to be optimal.

No 2-identifying code in the hexagonal grid can have a density less than 4/19.

The previous lower bound of 1/5 is surpassed by the new optimality result.

Abstract

An $r$-identifying code in a graph $G = (V,E)$ is a subset $C \subseteq V$ such that for each $u \in V$ the intersection of $C$ and the ball of radius $r$ centered at $u$ is non-empty and unique. Previously, $r$-identifying codes have been studied in various grids. In particular, it has been shown that there exists a 2-identifying code in the hexagonal grid with density 4/19 and that there are no 2-identifying codes with density smaller than 2/11. Recently, the lower bound has been improved to 1/5 by Martin and Stanton (2010). In this paper, we prove that the 2-identifying code with density 4/19 is optimal, i.e. that there does not exist a 2-identifying code in the hexagonal grid with smaller density.