Optimal radio labelings of the Cartesian product of the generalized Peterson graph and tree
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
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2023
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| _version_ | 1866914749730521088 |
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| author | Vasoya, Payal Bantva, Devsi |
| author_facet | Vasoya, Payal Bantva, Devsi |
| contents | A radio labeling of a graph $G$ is a function $f : V(G) \rightarrow \{0,1,2,\ldots\}$ such that $|f(u)-f(v)| \geq diam(G) + 1 - d(u,v)$ for every pair of distinct vertices $u,v$ of $G$. The radio number of $G$, denoted by $rn(G)$, is the smallest number $k$ such that $G$ has radio labeling $f$ with max$\{f(v):v \in V(G)\} = k$. In this paper, we give a lower bound for the radio number for the Cartesian product of the generalized Petersen graph and tree. We present two necessary and sufficient conditions, and three other sufficient conditions to achieve the lower bound. Using these results, we determine the radio number for the Cartesian product of the Peterson graph and stars. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2304_10094 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Optimal radio labelings of the Cartesian product of the generalized Peterson graph and tree Vasoya, Payal Bantva, Devsi Combinatorics Discrete Mathematics 05C78, 05C15 A radio labeling of a graph $G$ is a function $f : V(G) \rightarrow \{0,1,2,\ldots\}$ such that $|f(u)-f(v)| \geq diam(G) + 1 - d(u,v)$ for every pair of distinct vertices $u,v$ of $G$. The radio number of $G$, denoted by $rn(G)$, is the smallest number $k$ such that $G$ has radio labeling $f$ with max$\{f(v):v \in V(G)\} = k$. In this paper, we give a lower bound for the radio number for the Cartesian product of the generalized Petersen graph and tree. We present two necessary and sufficient conditions, and three other sufficient conditions to achieve the lower bound. Using these results, we determine the radio number for the Cartesian product of the Peterson graph and stars. |
| title | Optimal radio labelings of the Cartesian product of the generalized Peterson graph and tree |
| topic | Combinatorics Discrete Mathematics 05C78, 05C15 |
| url | https://arxiv.org/abs/2304.10094 |