The k-Sudoku Number of Graphs
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866913826250686464 |
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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 $χ(G)$. Let $ k \geq χ(G) $ and $S \subseteq V$. 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 $k$- Sudoku coloring of $G$, if $C_0$ can be uniquely extended to a $k$-coloring of $G$. The smallest order of such an induced subgraph $G[S]$ of $G$ which admits a $k$- Sudoku coloring is called $k$- Sudoku number of $G$ and is denoted by $sn(G,k)$. When $k=χ(G)$, we call $k$- Sudoku number of $G$ as Sudoku number of $G$ and is denoted by $sn(G)$. In this paper, we have obtained the $3$- Sudoku number of some bipartite graphs $P_n$, $C_{2n}$, $K_{m,n}$, $B_{m,n}$ and $G \circ lK_1$, where $G$ is a bipartite graph and $l\geq1$. Also, we have obtained the necessary and sufficient conditions for a bipartite graph $G$ to have $sn(G,3)$ equal to $n$, $n-1$ or $n-2$. Also, we study the relation between $k$- Sudoku number of a graph $G$ and the Sudoku number of a supergraph $H$ of $G$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_04920 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The k-Sudoku Number of Graphs Nair, Manju S S, Aparna Lakshmanan Arumugam, S Combinatorics Primary: 05C15, Secondary: 05C76 Let $G=(V,E)$ be a graph of order $n$ with chromatic number $χ(G)$. Let $ k \geq χ(G) $ and $S \subseteq V$. 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 $k$- Sudoku coloring of $G$, if $C_0$ can be uniquely extended to a $k$-coloring of $G$. The smallest order of such an induced subgraph $G[S]$ of $G$ which admits a $k$- Sudoku coloring is called $k$- Sudoku number of $G$ and is denoted by $sn(G,k)$. When $k=χ(G)$, we call $k$- Sudoku number of $G$ as Sudoku number of $G$ and is denoted by $sn(G)$. In this paper, we have obtained the $3$- Sudoku number of some bipartite graphs $P_n$, $C_{2n}$, $K_{m,n}$, $B_{m,n}$ and $G \circ lK_1$, where $G$ is a bipartite graph and $l\geq1$. Also, we have obtained the necessary and sufficient conditions for a bipartite graph $G$ to have $sn(G,3)$ equal to $n$, $n-1$ or $n-2$. Also, we study the relation between $k$- Sudoku number of a graph $G$ and the Sudoku number of a supergraph $H$ of $G$. |
| title | The k-Sudoku Number of Graphs |
| topic | Combinatorics Primary: 05C15, Secondary: 05C76 |
| url | https://arxiv.org/abs/2505.04920 |