Abstract. 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
$w\,\in \,W$
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
$\dim\left( G \right)$
. The circulant graph
${{C}_{n}}\left( 1,\,2,\,.\,.\,.\,,\,t \right)$
consists of the vertices
${{v}_{0}},\,{{v}_{1\,}},\,.\,.\,.\,,{{v}_{n\,-\,1}}$
and the edges
${{v}_{i}}{{v}_{i\,+\,j}}$
, where
$0\,\le \,i\,\le \,n\,-\,1,1\,\le \,j\,\le \,t\,\left( 2\,\le \,t\,\le \,\left\lfloor \frac{n}{2} \right\rfloor \right)$
, the indices are taken modulo
$n$
. Grigorious, Manuel, Miller, Rajan, and Stephen proved that
$\dim\left( {{C}_{n}}\left( 1,\,2,\,.\,.\,.\,,\,t \right) \right)\,\ge \,t\,+\,1$
for
$t\,<\,\left\lfloor \frac{n}{2} \right\rfloor ,\,n\,\ge \,3$
, and they presented a conjecture saying that
$\dim\left( {{C}_{n}}\left( 1,\,2,\,.\,.\,.\,,\,t \right) \right)\,=\,t\,+\,p\,-\,1$
for
$n\,=\,2tk\,+\,t\,+\,p$
, where
$3\,\le \,p\,\le \,t\,+\,1$
. We disprove both statements. We show that if
$t\,\ge \,4$
is even, there exists an infinite set of values of
$n$
such that
$\dim\left( {{C}_{n}}\left( 1,\,2,\,.\,.\,.\,,t \right) \right)\,=\,t$
. We also prove that
$\dim\left( {{C}_{n}}\left( 1,\,2,\,.\,.\,.\,,\,t \right) \right)\,\le \,t\,+\,\frac{p}{2}$
for
$n\,=\,2tk\,+\,t\,+\,p$
, where
$t$
and
$p$
are even,
$t\,\ge \,4,\,2\,\le \,p\,\le \,t$
, and
$k\,\ge \,1$
.