Optimal identifying codes of two families of Cayley graphs

Abstract

An identifying code of a graph Γ is a subset C of the vertex set V such that for each x ∈ V , the intersection of its closed neighbourhood with C is nonempty and unique. If Γ is a finite graph, the density of an identifying code C is defined as | C | | V | , which naturally extends to a definition of density in certain infinite graphs which are locally finite. Denote by d ∗ ( Γ ) the infimum of the density of an identifying code of a finite or infinite graph Γ . In this paper, we study identifying codes of an infinite Cayley graph Γ n , which is the Cartesian product of an infinite path and a complete graph on n vertices. We prove that d ∗ ( Γ 3 ) = 5 12 , d ∗ ( Γ 4 ) = 9 22 and d ∗ ( Γ n ) = n − 1 2 n for n ≥ 5 . As an application, we obtain d ∗ ( Γ ) if Γ is a connected quintic Cayley graph over a generalized quaternion group.