Let $f$ be a proper $k$-coloring of a connected graph $G$ and $Pi=(V_1,V_2,ldots,V_k)$ be an ordered partition of $V(G)$ into the resulting color classes. For a vertex $v$ of $G$, the color code of $v$ with respect to $Pi$ ...

For a graph $G,$ let $mathscr{D}(G)$ be the set of all strong digraphs $D$ obtained by the orientations of $G.$ The orientation number of $G$ is $stackrel{rightarrow}{d}!!(G),=$ min ${d(D),|,D,in,mathscr{D}(G)},$ ...