on odd-graceful coloring of graphs

For a graph $g(v,e)$ which is undirected, simple, and finite, we denote by $|v|$ and $|e|$ the cardinality of the vertex set $v$ and the edge set $e$ of $g$, respectively. a textit{graceful labeling} $f$ for the graph $g$ is an injective function ${f}:vrightarrow {0,1,2,..., |e|}$ such that ${|f(u)-f(v)|:uvin e}={1,2,...,|e|}$. a graph that has a graceful-labeling is called textit{graceful} graph. a vertex (resp. edge) coloring is an assignment of color (positive integer) to every vertex (resp. edge) of $g$ such that any two adjacent vertices (resp. edges) have different colors. a textit{graceful coloring} of $g$ is a vertex coloring $c: vrightarrow {1,2,ldots, k},$ for some positive integer $k$, which induces edge coloring $|c(u)-c(v)|$, $uvin e$. if $c$ also satisfies additional property that every induced edge color is odd, then the coloring $c$ is called an textit{odd-graceful coloring} of $g$. if an odd-graceful coloring $c$ exists for $g$, then the smallest number $k$ which maintains $c$ as an odd-graceful coloring, is called textit{odd-graceful chromatic number} for $g$. in the latter case we will denote the odd-graceful chromatic number of $g$ as $mathcal{x}_{og}(g)=k$. otherwise, if $g$ does not admit odd-graceful coloring, we will denote its odd-graceful chromatic number as $mathcal{x}_{og}(g)=infty$. in this paper, we derived some facts of odd-graceful coloring and determined odd-graceful chromatic numbers of some basic graphs.