On the metric dimension of circulant graphs

In this note, we bound the metric dimension of the circulant graphs C-n(1, 2,..., t). We shall prove that if n = 2tk + t and if t is odd, then dim(C-n(1, 2,..., t)) = t + 1, which confirms Conjecture 4.1.1 in Chau and Gosselin (2017, Opuscula Mathematica 37, 509-534). In Vetrik (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,..., t)) <= t + inverted right perpendicularp/2inverted left perpendicular for n = 2tk + t + p, where t >= 4 is even, 1 <= p <= t + 1, and k >= 1. Inspired by his work, we show that dim(C-n(1, 2,..., t)) <= t + inverted right perpendicularp/2inverted left perpendicular for n = 2tk + t + p, where t >= 5 is odd, 2 <= p <= t + 1, and k >= 2.