On the Metric Dimension of Circulant Graphs

Pub Date : 2023-09-28 DOI:10.4153/s0008439523000759

Rui Gao, Yingqing Xiao, Zhanqi Zhang

引用次数: 0

Abstract

Abstract 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 $k\geq 2$ .

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关于循环图的度量维数

摘要本文对循环图$C_n(1,2,\ldots ,t)$的度量维数进行了定界。我们将证明，如果$n=2tk+t$和如果t是奇数，则$\dim (C_n(1,2,\ldots ,t))=t+1$，这证实了Chau和Gosselin (2017, Mathematica 37, 509-534)的猜想4.1.1。见Vetrík(2017，加拿大数学通报60,206-216;2020，《数学讨论》。图论40,67-76)，作者已经证明$\dim (C_n(1,2,\ldots ,t))\leq t+\left \lceil \frac {p}{2}\right \rceil $对于$n=2tk+t+p$，其中$t\geq 4$是偶数，$1\leq p\leq t+1$，和$k\geq 1$。受他的工作启发，我们展示了$\dim (C_n(1,2,\ldots ,t))\leq t+\left \lfloor \frac {p}{2}\right \rfloor $代表$n=2tk+t+p$，其中$t\geq 5$是奇数，$2\leq p\leq t+1$和$k\geq 2$。

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