Geodesic
On the metric dimension of circulant graphs
Canadian mathematical bulletin, Tome 67 (2024) no. 2, pp. 328-337

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DOI

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 $k\geq 2$.
DOI : 10.4153/S0008439523000759
Mots-clés : Metric dimension, resolving set, circulant graph, distance 
Gao, Rui; Xiao, Yingqing; Zhang, Zhanqi. On the metric dimension of circulant graphs. Canadian mathematical bulletin, Tome 67 (2024) no. 2, pp. 328-337. doi: 10.4153/S0008439523000759
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