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On the metric dimension of circulant graphs

Published online by Cambridge University Press:  28 September 2023

Rui Gao
School of Statistics and Mathematics, Shandong University of Finance and Economics, Jinan, Shandong 250014, China e-mail:
Yingqing Xiao
School of Mathematics, Hunan University, Changsha, Hunan 410082, China e-mail:
Zhanqi Zhang*
School of Mathematics and Computational Science, Hunan University of Science and Technology, Xiangtan, Hunan 411201, China


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 $k\geq 2$.

© The Author(s), 2023. Published by Cambridge University Press on behalf of The Canadian Mathematical Society

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This work was supported by the Natural Science Foundation of Shandong Province (Grant No. ZR2021QG036).


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