The partition dimension of Cayley digraphs

Let G be a (di)graph and S a set of vertices of G. We say S resolves two vertices u and v of G if d(u, S) 6= d(v, S). A partition Π = {P1, P2, . . . , Pk} of V (G) is a resolving partition of G if every two vertices of G are resolved by Pi for some i (1 ≤ i ≤ k). The smallest cardinality of a resolving partition of G, denoted by pd(G), is called the partition dimension of G. A vertex r of G resolves a pair u, v of vertices of G if d(u, r) 6= d(v, r). A set R of vertices of G is a resolving set for G if every two vertices of G are resolved by some vertex of R. The smallest cardinality of a resolving set of vertices, denoted by dim(G), is called the metric dimension of G. We begin by disproving a conjecture made by Chartrand, Salehi and Zhang regarding the partition dimension of products of graphs. The partition dimension of Cayley digraphs of abelian groups with a specific minimal set of generators is shown to be at most one more than the number of generators with equality for one or two generators. It is known that pd(G) ≤ dim(G)+1. It is pointed out that for every positive integer M there are Cayley digraphs D for which dim(D) − pd(D) ≥ M , and that there are classes of Cayley digraphs D such that pd(D) dim(D) → 0 as |V (D)| → ∞. Moreover, it is shown that the partition dimension of the Cayley