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

Authors
  • Junnila, Ville
  • Laihonen, Tero
Type
Preprint
Publication Date
Feb 03, 2012
Submission Date
Feb 03, 2012
Identifiers
arXiv ID: 1202.0670
Source
arXiv
License
Yellow
External links

Abstract

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

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