The k-Sudoku Number of Graphs

Let $G=(V,E)$ be a graph of order $n$ with chromatic number $\chi(G)$. Let $k \geq \chi(G)$ 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 $k=\chi(G)$, 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 $G$.