Intertwining local (adjacency) metric dimension with the clique number of a graph
Fuente:
arXiv
Enregistré dans:
| Auteurs principaux: | , , |
|---|---|
| Format: | Preprint |
| Publié: |
2025
|
| Sujets: | |
| Accès en ligne: | |
| Tags: |
Ajouter un tag
Pas de tags, Soyez le premier à ajouter un tag!
|
| _version_ | 1866915398073450496 |
|---|---|
| author | Ghalavand, Ali Klavžar, Sandi Li, Xueliang |
| author_facet | Ghalavand, Ali Klavžar, Sandi Li, Xueliang |
| contents | Let $G$ be a simple connected graph with order $ n(G)$, local metric dimension $ {\rm dim}_l(G)$, local adjacency metric dimension $ {\rm dim}_{A,l}(G)$, and clique number $ ω(G)$, where $G\not\cong K_{n(G)}$ and $ω(G)\geq3$. It is proved that $ {\rm dim}_{A,l}(G) \leq \left\lfloor \left(\frac{ω(G) - 2}{ω(G) - 1}\right)n(G)\right\rfloor$. Consequently, the conjecture asserting that the latter expression is an upper bound for ${\rm dim}_l(G)$ is confirmed. It is important to note that there are infinitely many graphs that satisfy the equalities. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_13777 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Intertwining local (adjacency) metric dimension with the clique number of a graph Ghalavand, Ali Klavžar, Sandi Li, Xueliang Combinatorics Let $G$ be a simple connected graph with order $ n(G)$, local metric dimension $ {\rm dim}_l(G)$, local adjacency metric dimension $ {\rm dim}_{A,l}(G)$, and clique number $ ω(G)$, where $G\not\cong K_{n(G)}$ and $ω(G)\geq3$. It is proved that $ {\rm dim}_{A,l}(G) \leq \left\lfloor \left(\frac{ω(G) - 2}{ω(G) - 1}\right)n(G)\right\rfloor$. Consequently, the conjecture asserting that the latter expression is an upper bound for ${\rm dim}_l(G)$ is confirmed. It is important to note that there are infinitely many graphs that satisfy the equalities. |
| title | Intertwining local (adjacency) metric dimension with the clique number of a graph |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2507.13777 |