The Metric Dimension of Circulant Graphs and Their Cartesian Products

Let G = (V,E) be a connected graph (or hypergraph) and let d(x, y) denote the distance between vertices x, y ∈ V (G). A subset W ⊆ V (G) is called a resolving set for G if for every pair of distinct vertices x, y ∈ V (G), there is w ∈ W such that d(x,w) 6= d(y, w). The minimum cardinality of a resolving set for G is called the metric dimension of G, denoted by β(G). The circulant graph Cn(1, 2, . . . , t) has vertex set {v0, v1, . . . , vn−1} and edges vivi+j where 0 ≤ i ≤ n − 1 and 1 ≤ j ≤ t and the indices are taken modulo n (2 ≤ t ≤ ⌊ n 2 ⌋ ). In this paper we determine the exact metric dimension of the circulant graphs Cn(1, 2, . . . , t), extending previous results due to Borchert and Gosselin (2013), Grigorious et al. (2014), and Vetrík (2016). In particular, we show that β(Cn(1, 2, . . . , t)) = β(Cn+2t(1, 2, . . . , t)) for large enough n, which implies that the metric dimension of these circulants is completely determined by the congruence class of n modulo 2t. We determine the exact value of β(Cn(1, 2, . . . , t)) for n ≡ 2 mod 2t and n ≡ (t+ 1) mod 2t and we give better bounds on the metric dimension of these circulants for n ≡ 0 mod 2t and n ≡ 1 mod 2t. In addition, we bound the metric dimension of Cartesian products of circulant graphs.