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 ${f}:V\rightarrow \{0,1,2,..., |E|\}$ such that $\{|f(u)-f(v)|:uv\in E\}=\{1,2,...,|E|\}$. 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 $c: V\rightarrow \{1,2,\ldots, k\},$ for some positive integer $k$, which induces edge coloring $|c(u)-c(v)|$, $uv\in E$. 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.