The connection between the magical coloring of trees

Let $f$ be a set-ordered edge-magic labeling of a graph $G$ from $V(G)$ and $E(G)$ to $[0,p-1]$ and $[1,p-1]$, respectively; it also satisfies the following conditions: $|f(V(G))|=p$, $maxf(X)<minf(Y)$, and $f(x)+f(y)+f(xy)=C$ for each edge $xy\in E(G)$. In this paper, we removed the restriction that the labeling of vertices could not be repeated, and presented the concept of magical colorings including edge-magic coloring, edge-difference coloring, felicitous-difference coloring, and graceful-difference coloring. We studied the magical colorings on the tree and proved the existence of four kinds of magical colorings on the tree from a set-ordered edge-magic labeling. Further, we revealed the transformation relationship between these kinds of colorings.