New Bounds on the Minimum Density of a Vertex Identifying Code for the Infinite Hexagonal Grid

TL;DR

This paper improves the bounds on the minimum density of vertex identifying codes in the infinite hexagonal grid, presenting three new codes with density 3/7 and raising the lower bound to 5/12, narrowing the gap in known values.

Contribution

The paper introduces three new vertex identifying codes with density 3/7 and enhances the lower bound to 5/12 for the infinite hexagonal grid.

Findings

New codes with density 3/7 are constructed.

Lower bound on density improved to 5/12.

The bounds on minimum density are tightened.

Abstract

For a graph, $G$, and a vertex $v \in V(G)$, let $N[v]$ be the set of vertices adjacent to and including $v$. A set $D \subseteq V(G)$ is a vertex identifying code if for any two distinct vertices $v_1, v_2 \in V(G)$, the vertex sets $N[v_1] \cap D$ and $N[v_2] \cap D$ are distinct and non-empty. We consider the minimum density of a vertex identifying code for the infinite hexagonal grid. In 2000, Cohen et al. constructed two codes with a density of $3/7 \approx 0.428571$, and this remains the best known upper bound. Until now, the best known lower bound was $12/29 \approx 0.413793$ and was proved by Cranston and Yu in 2009. We present three new codes with a density of 3/7, and we improve the lower bound to $5/12 \approx 0.416667$.