On the local metric dimension of $K_4$-free graphs
Fuente:
arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866912405889482752 |
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| author | Ghalavand, Ali Klavžar, Sandi Li, Xueliang |
| author_facet | Ghalavand, Ali Klavžar, Sandi Li, Xueliang |
| contents | Let $G$ be a graph of order $ n(G) $, local metric dimension $ \dim_l(G) $, and clique number $ ω(G) $. It has been conjectured that if $ n(G) \geq ω(G) + 1 \geq 4 $, then $ \dim_l(G) \leq \left( \frac{ω(G) - 2}{ω(G) - 1} \right) n(G) $. In this paper the conjecture is confirmed for the case $ ω(G) = 3 $. Consequently, a problem regarding the local metric dimension of planar graphs is also resolved. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_00414 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the local metric dimension of $K_4$-free graphs Ghalavand, Ali Klavžar, Sandi Li, Xueliang Combinatorics Let $G$ be a graph of order $ n(G) $, local metric dimension $ \dim_l(G) $, and clique number $ ω(G) $. It has been conjectured that if $ n(G) \geq ω(G) + 1 \geq 4 $, then $ \dim_l(G) \leq \left( \frac{ω(G) - 2}{ω(G) - 1} \right) n(G) $. In this paper the conjecture is confirmed for the case $ ω(G) = 3 $. Consequently, a problem regarding the local metric dimension of planar graphs is also resolved. |
| title | On the local metric dimension of $K_4$-free graphs |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2506.00414 |