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

: Junnila V, Laihonen T

Publisher: ELECTRONIC JOURNAL OF COMBINATORICS

: 2012

: The Electronic Journal of Combinatorics

: ELECTRONIC JOURNAL OF COMBINATORICS

: ELECTRON J COMB

: 2

: 19

: 2

: 16

: 1077-8926

An r-identifying code in a graph G = (V, E) is a subset C subset of V such that for each u is an element of 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.