The metric dimension of circulant graphs

Abstract

A pair of vertices $x$ and $y$ in a graph $G$ are said to be resolved by a vertex $w$ if the distance from $x$ to $w$ is not equal to the distance from $y$ to $w$. We say that $G$ is resolved by a subset of its vertices $W$ if every pair of vertices in $G$ is resolved by some vertex in $W$. The minimum cardinality of a resolving set for $G$ is called the metric dimension of $G$, denoted by $\dim(G)$. The circulant graph $C_n(1,2,\ldots,t)$ is the Cayley graph $ay(\mathbb{Z}_n:\{\pm 1, \pm 2, \ldots, \pm t\})$. In this note we prove that, for $n=2kt+2t$, $\dim(C_n(1,2,\ldots,t))\geq t+2$, confirming Conjecture 4.1.2 in [K. Chau, S. Gosselin, The metric dimension of circulant graphs and their Cartesian products, Opuscula Math. 37 (2017), 509-534].