Sharp lower bounds on the metric dimension of circulant graphs

Knor, Martin; Škrekovski, Riste; Vetrík, Tomáš

doi:10.22049/cco.2023.28792.1725

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	Sharp lower bounds on the metric dimension of circulant graphs
Communications in Combinatorics and Optimization
مقاله 5، دوره 10، شماره 1، خرداد 2025، صفحه 79-98 اصل مقاله (407.61 K)
نوع مقاله: Original paper
شناسه دیجیتال (DOI): 10.22049/cco.2023.28792.1725
نویسندگان
Martin Knor¹؛ Riste Škrekovski²^{، 3}؛ Tomáš Vetrík^* ⁴
¹Slovak University of Technology, Bratislava, Slovakia
²Faculty of Mathematics and Physics, University of Ljubljana, Ljubljana, Slovenia
³Faculty of Information Studies, Novo Mesto, Slovenia
⁴Department of Mathematics and Applied Mathematics, University of the Free State, Bloemfontein, South Africa
چکیده
For $n \ge 2t+1$ where $t \ge 1$, the circulant graph $C_n (1, 2, \dots , t)$ consists of the vertices $v_0, v_1, v_2, \dots , v_{n-1}$ and the edges $v_i v_{i+1}$, $v_i v_{i+2}, \dots , v_i v_{i + t}$, where $i = 0, 1, 2, \dots , n-1$, and the subscripts are taken modulo $n$. We prove that the metric dimension ${\rm dim} (C_n (1, 2, \dots , t)) \ge \left\lceil \frac{2t}{3} \right\rceil + 1$ for $t \ge 5$, where the equality holds if and only if $t = 5$ and $n = 13$. Thus ${\rm dim} (C_n (1, 2, \dots , t)) \ge \left\lceil \frac{2t}{3} \right\rceil + 2$ for $t \ge 6$. This bound is sharp for every $t \ge 6$.
کلیدواژه‌ها
Cayley graph؛ distance؛ resolving set

مراجع
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Sharp lower bounds on the metric dimension of circulant graphs