On the metric dimension of circulant graphs

Abstract

Abstract In this note, we bound the metric dimension of the circulant graphs $C_n(1,2,\ldots ,t)$ . We shall prove that if $n=2tk+t$ and if t is odd, then $\dim (C_n(1,2,\ldots ,t))=t+1$ , which confirms Conjecture 4.1.1 in Chau and Gosselin (2017, Opuscula Mathematica 37, 509–534). In Vetrík (2017, Canadian Mathematical Bulletin 60, 206–216; 2020, Discussiones Mathematicae. Graph Theory 40, 67–76), the author has shown that $\dim (C_n(1,2,\ldots ,t))\leq t+\left \lceil \frac {p}{2}\right \rceil $ for $n=2tk+t+p$ , where $t\geq 4$ is even, $1\leq p\leq t+1$ , and $k\geq 1$ . Inspired by his work, we show that $\dim (C_n(1,2,\ldots ,t))\leq t+\left \lfloor \frac {p}{2}\right \rfloor $ for $n=2tk+t+p$ , where $t\geq 5$ is odd, $2\leq p\leq t+1$ , and $k\geq 2$ .