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A rooted tree $\vec{R}$ is a rooted subtree of a tree $T$ if the treeobtained by replacing the directed edges of $\vec{R}$ by undirected edges is asubtree of $T$. We study the problem of assigning minimum number of colors to agiven set of rooted subtrees $\mathcal{R}$ of a given tree $T$ such that if anytwo rooted subtrees share a directed edge, then they are assigned differentcolors. The problem is NP hard even in the case when the degree of $T$ isrestricted to $3$. We present a $\frac{5}{2}$-approximation algorithm for thisproblem. The motivation for studying this problem stems from the problem ofassigning wavelengths to multicast traffic requests in all-optical WDM treenetworks.

A $\frac{5}{2}$-Approximation Algorithm for Coloring Rooted Subtrees of a Degree $3$ Tree