On the metric dimension of circulant graphs

In this note, we bound the metric dimension of the circulant graphs Cₙ(1,2,… ,t) . We shall prove that if $n=2tk+t$ and if t is odd, then (Cₙ(1,2,… ,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 (Cₙ(1,2,… ,t))≤ t+ p/2 for $n=2tk+t+p$ , where t≥ 4 is even, 1≤ p≤ t+1 , and k≥ 1 . Inspired by his work, we show that (Cₙ(1,2,… ,t))≤ t+ p/2 for $n=2tk+t+p$ , where t≥ 5 is odd, 2≤ p≤ t+1 , and k≥ 2 .