Density of identifying codes of hexagonal grids with finite number of rows

In a graph G , a set C ⊆ V ( G ) is an identifying code if, for all vertices v in G , the sets N [ v ] ∩ C are all nonempty and pairwise distinct, where N [ v ] denotes the closed neighbourhood of v . We focus on the minimum density of identifying codes of infinite hexagonal grids H k with k rows, denoted by d * ( H k ), and present optimal solutions for k ≤ 5. Using the discharging method, we also prove a lower bound in terms of maximum degree for the minimum-density identifying codes of well-behaved infinite graphs. We prove that d * ( H 2 ) = 9/20, d * ( H 3 ) = 6/13 ≈ 0.4615, d * ( H 4 ) = 7/16 = 0.4375 and d * ( H 5 ) = 11/25 = 0.44. We also prove that H 2 has a unique periodic identifying code with minimum density.