Interplay between the local metric dimension and the clique number of a graph
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arXiv
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| Natura: | Preprint |
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2024
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| _version_ | 1866912166279380992 |
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| author | Ghalavand, Ali Klavžar, Sandi Li, Xueliang |
| author_facet | Ghalavand, Ali Klavžar, Sandi Li, Xueliang |
| contents | The local metric dimension ${\rm dim}_l$ in relation to the clique number $ω$ is investigated. It is proved that if $ω(G)\leq n(G)-3$, then ${\rm dim}_l(G) \leq n(G)-3$ and the graphs attaining the bound classified. Moreover, the graphs $G$ with ${\rm dim}_l(G) = n(G)-3$ are listed (with no condition on the clique number). It is proved that if $ω(G)=n(G)-2$, then $n(G)-4 \leq {\rm dim}_l(G)\leq n(G)-3$, and all graphs are divided into two groups depending on which of the options applies. The conjecture asserting that for any graph $G$ we have ${\rm dim}_l(G) \leq \left[(ω(G)-2)/(ω(G)-1)\right] \cdot n(G)$ is proved for all graphs with $ω(G)\in\{n(G)-1,n(G)-2,n(G)-3\}$. A negative answer is given for the problem whether every planar graph fulfills the inequality ${\rm dim}_l(G) \leq \lceil (n(G)+1)/2 \rceil$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_17074 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Interplay between the local metric dimension and the clique number of a graph Ghalavand, Ali Klavžar, Sandi Li, Xueliang Combinatorics The local metric dimension ${\rm dim}_l$ in relation to the clique number $ω$ is investigated. It is proved that if $ω(G)\leq n(G)-3$, then ${\rm dim}_l(G) \leq n(G)-3$ and the graphs attaining the bound classified. Moreover, the graphs $G$ with ${\rm dim}_l(G) = n(G)-3$ are listed (with no condition on the clique number). It is proved that if $ω(G)=n(G)-2$, then $n(G)-4 \leq {\rm dim}_l(G)\leq n(G)-3$, and all graphs are divided into two groups depending on which of the options applies. The conjecture asserting that for any graph $G$ we have ${\rm dim}_l(G) \leq \left[(ω(G)-2)/(ω(G)-1)\right] \cdot n(G)$ is proved for all graphs with $ω(G)\in\{n(G)-1,n(G)-2,n(G)-3\}$. A negative answer is given for the problem whether every planar graph fulfills the inequality ${\rm dim}_l(G) \leq \lceil (n(G)+1)/2 \rceil$. |
| title | Interplay between the local metric dimension and the clique number of a graph |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2412.17074 |