Sudoku Number of Corona of Graphs
Fuente:
arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866912232504295424 |
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| author | Nair, Manju S S, Aparna Lakshmanan Arumugam, S |
| author_facet | Nair, Manju S S, Aparna Lakshmanan Arumugam, S |
| contents | Let $G = (V,E)$ be a graph of order $n$ with chromatic number $χ(G) = k$, let $S \subset V$ and let $C_0$ be a $k$-coloring of the induced subgraph $G[S]$. The coloring $C_0$ is called an extendable coloring, if $C_0$ can be extended to a $k$-coloring of $G$ and it is a Sudoku coloring of $G$ if the extension is unique. The smallest order of such an induced subgraph $G[S]$ of $G$ which admits a Sudoku coloring is called the Sudoku number of $G$ and is denoted by $sn(G)$. In this paper, we first introduce the notion of uniquely color extendable vertex and then we obtain the lower and upper bounds for the Sudoku number of $G \circ K_1$. Some families of graphs which attain these bounds are also obtained. The exact value of the Sudoku number of corona of $C_n$, $W_n$ and $K_n$ with $K_1$ and $C_n \circ P_m$ are also obtained. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_08933 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Sudoku Number of Corona of Graphs Nair, Manju S S, Aparna Lakshmanan Arumugam, S Combinatorics 05C15, 05C76 Let $G = (V,E)$ be a graph of order $n$ with chromatic number $χ(G) = k$, let $S \subset V$ and let $C_0$ be a $k$-coloring of the induced subgraph $G[S]$. The coloring $C_0$ is called an extendable coloring, if $C_0$ can be extended to a $k$-coloring of $G$ and it is a Sudoku coloring of $G$ if the extension is unique. The smallest order of such an induced subgraph $G[S]$ of $G$ which admits a Sudoku coloring is called the Sudoku number of $G$ and is denoted by $sn(G)$. In this paper, we first introduce the notion of uniquely color extendable vertex and then we obtain the lower and upper bounds for the Sudoku number of $G \circ K_1$. Some families of graphs which attain these bounds are also obtained. The exact value of the Sudoku number of corona of $C_n$, $W_n$ and $K_n$ with $K_1$ and $C_n \circ P_m$ are also obtained. |
| title | Sudoku Number of Corona of Graphs |
| topic | Combinatorics 05C15, 05C76 |
| url | https://arxiv.org/abs/2402.08933 |