Sharp lower bounds on the metric dimension of circulant graphs

Knor, Martin (author), Škrekovski, Riste (author), Vetrík, Tomáš (author)

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 $t \ge 6$.

• Language: English
• Keywords: Cayley graph, distance, resolving set
• Work type: Article
• Typology: 1.01 - Original Scientific Article
• Organization: FMF - Faculty of Mathematics and Physics
• Publication status: Published
• Publication version: Version of Record
• Publication date: 01.01.2025
• Year: 2025
• Number of pages: Str. 79-98
• Numbering: Vol. 10, no. 1
• PID: 20.500.12556/RUL-168563
• UDC: 519.17
• ISSN on article: 2538-2128
• DOI: 10.22049/cco.2023.28792.1725
• COBISS.SI-ID: 232767747
• Publication date in RUL: 17.04.2025

Record is a part of a journal

• Title: Communications in combinatorics and optimization
• Shortened title: Commun. comb. optim.
• Publisher: Azarbaijan Shahid Madani University
• ISSN: 2538-2128
• COBISS.SI-ID: 23159048

Licences

• License: CC BY-SA 4.0, Creative Commons Attribution-ShareAlike 4.0 International
• Link: http://creativecommons.org/licenses/by-sa/4.0/
• Description: This Creative Commons license is very similar to the regular Attribution license, but requires the release of all derivative works under this same license.

Secondary language

• Language: Slovenian
• Keywords: Cayleyev graf, razdalja