A New Lower Bound on the Density of Vertex Identifying Codes for the Infinite Hexagonal Grid

TL;DR

This paper improves the lower bound on the density of vertex identifying codes in the infinite hexagonal grid from approximately 0.410256 to 0.413793, narrowing the gap between known bounds.

Contribution

It establishes a new lower bound of 12/29 for the density of identifying codes in the infinite hexagonal grid, advancing the understanding of optimal code densities.

Findings

New lower bound of 12/29 for code density

Previous bounds were 16/39 and 3/7

The result tightens the known range of code densities

Abstract

Given a graph $G$, an identifying code $C \subseteq V(G)$ is a vertex set such that for any two distinct vertices $v_1,v_2\in V(G)$, the sets $N[v_1]\cap C$ and $N[v_2]\cap C$ are distinct and nonempty (here $N[v]$ denotes a vertex $v$ and its neighbors). We study the case when $G$ is the infinite hexagonal grid $H$. Cohen et.al. constructed two identifying codes for $H$ with density $3/7$ and proved that any identifying code for $H$ must have density at least $16/39\approx0.410256$. Both their upper and lower bounds were best known until now. Here we prove a lower bound of $12/29\approx0.413793$.