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

Florent Foucaud (1) , Tero Laihonen (2) , Aline Parreau (3)

1 LaBRI - Laboratoire Bordelais de Recherche en Informatique
2 Turku Centre for Computer Science
3 IF - Institut Fourier

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 $\frac{3}{7}$ and that there are no such identifying codes with density smaller than $\frac{5}{12}$. 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 $\frac{47}{111}$.

Mots clés: Identifying codes; king grid; density; discharging method

Domaines: Combinatoire [math.CO]; Mathématique discrète [cs.DM]

Florent Foucaud, Tero Laihonen, Aline Parreau. An improved lower bound for (1,<=2)-identifying codes in the king grid. Advances in Mathematics of Communications, 2014, 8 (1), pp.35-52. ⟨10.3934/amc.2014.8.35⟩. ⟨hal-00639998⟩