The Metric Dimension of Circulant Graphs

A subset W of the vertex set of a graph G is called a resolving set of G if for every pair of distinct vertices u,v of G, there is w is an element of W such that the distance of w and u is different from the distance of w and v. The cardinality of a smallest resolving set is called the metric dimension of G, denoted by dim(G). The circulant graph C-n (1, 2,..., t) consists of the vertices v0,v1..... vn-1 and the edges vivi+j, where 0 <= i <= n-1,1 <= j <= t (2 < t < L[n/2]), the indices are taken modulo n. Grigorious, Manuel, Miller, Rajan, and Stephen proved that dim(Cn (1, 2,...., t)) >=_ t + 1 for t <[n/2],n >= 3, and they presented a conjecture saying that dim(Cn (1, 2,..., t)) = t + p -1 for n = 2tk + t + p, where 3 < p < t + 1. We disprove both statements. We show that if t 4 is even, there exists an infinite set of values of n such that dim( Cn (1, 2,..., t)) = t+p. We also prove that dim(Cn (1, 2,... t))<t+p/2 for n = 2tk + t + p, where t and p are even, t >= 4, 2 <= p <= t, and k >= 1.