Algorithm for l-vertex-coloring of trees

Let l be a positive integer, and G be a graph with nonnegative integer weights on edges. Then a generalized vertex-coloring, called an l-vertex-coloring of G, is an assignment of colors to the vertices in such a way that any two vertices u and v get different colors if the distance between u and v in G is at most l. A coloring is optimal if it uses minimum number of distinct colors. The l-vertex-coloring problem is to find an optimal l-vertex-coloring of a graph G. In this paper we present an O(n² + nΔ^l+1) time algorithm to find an l-vertex-coloring of a tree T, where Δ is the maximum degree of T. The algorithm takes O(n²) time if both l and Δ are bounded integers. We also compute the upper bound of colors to be 1+Δ ((Δ-1)^⌈1/2⌉-1)/(Δ-2)equations.