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

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 $12/29\approx0.413793$.

💡 Research Summary

The paper investigates vertex identifying codes on the infinite hexagonal grid (H) and improves the known lower bound on their density. An identifying code (C\subseteq V(H)) requires that for every vertex (v) the closed neighbourhood (N