Optimal identifying codes in the infinite 3-dimensional king grid

A subset C ⊆ V is an r -identifying code in a graph G = ( V , E ) if the sets I r ( v ) = { c ∈ C ∣ d ( c , v ) ≤ r } are distinct and non-empty for all vertices v ⊆ V . Here, d ( c , v ) denotes the number of edges on any shortest path from c to v . We consider the infinite n -dimensional king grid, i.e., the graph with vertex set V = Z n and the edge set E = { { x = ( x 1 , … , x n ) , y = ( y 1 , … , y n ) } ∣ | x i − y i | ≤ 1 for  i = 1 , … , n , x ≠ y } , and give some lower bounds on the density of an r -identifying code. In particular, we prove that for n = 3 and for all r ≥ 15 , the optimal density of an r -identifying code is 1 8 r 2 . The problem finding a minimum identifying code in the 3-dimensional king grid is equivalent with a minimum packing problem of cubes in the 3-dimensional lattice so that every point is covered by a distinct and non-empty subset of cubes.