An improved lower bound for (1,<=2)-identifying codes in the king grid

TL;DR

This paper improves the lower bound on the density of (1,<=2)-identifying codes in the king grid from 5/12 to 47/111, using a discharging method, advancing understanding of code density constraints.

Contribution

It introduces a new lower bound for the density of (1,<=2)-identifying codes in the king grid, refining previous bounds through a novel discharging technique.

Findings

Lower bound improved to 47/111

No (1,<=2)-identifying code can have density less than 47/111

Advances theoretical understanding of code density in grid graphs

Abstract

We call a subset $C$ of vertices of a graph $G$ a $(1,\leq \ell)$-identifying code if for all subsets $X$ of vertices with size at most $\ell$, the sets $\{c\in C |\exists u \in X, d(u,c)\leq 1\}$ are distinct. The concept of identifying codes was introduced in 1998 by Karpovsky, Chakrabarty and Levitin. Identifying codes have been studied in various grids. In particular, it has been shown that there exists a $(1,\leq 2)$-identifying code in the king grid with density 3/7 and that there are no such identifying codes with density smaller than 5/12. Using a suitable frame and a discharging procedure, we improve the lower bound by showing that any $(1,\leq 2)$-identifying code of the king grid has density at least 47/111.