  We introduce a new concept in graph coloring motivated by the popular Sudoku
puzzle. Let $G=(V,E)$ be a graph of order $n$ with chromatic number $\chi(G)=k$
and let $S\subseteq V.$ Let $\mathscr C_0$ be a $k$-coloring of the induced
subgraph $G[S].$ The coloring $\mathscr C_0$ is called an extendable coloring
if $\mathscr C_0$ can be extended to a $k$-coloring of $G.$ We say that
$\mathscr C_0$ is a Sudoku coloring of $G$ if $\mathscr C_0$ 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 Sudoku coloring is called the Sudoku
number of $G$ and is denoted by $sn(G).$ In this paper we initiate a study of
this parameter. We first show that this parameter is related to list coloring
of graphs. In Section 2, basic properties of Sudoku coloring that are related
to color dominating vertices, chromatic numbers and degree of vertices, are
given. Particularly, we obtained necessary conditions for $\mathscr C_0$ being
uniquely extendable, and for $\mathscr C_0$ being a Sudoku coloring. In Section
3, we determined the Sudoku number of various familes of graphs. Particularly,
we showed that a connected graph $G$ has $sn(G)=1$ if and only if $G$ is
bipartite. Consequently, every tree $T$ has $sn(T)=1$. Moreover, a graph $G$
with small chromatic number may have arbitrarily large Sudoku number.
Extendable coloring and Sudoku coloring are nice tools for providing a
$k$-coloring of $G$.
