Harmonious coloring of trees with large maximum degree

A harmonious coloring of G is a proper vertex coloring of G such that every pair of colors appears on at most one pair of adjacent vertices. The harmonious chromatic number of G, h(G), is the minimum number of colors needed for a harmonious coloring of G. We show that if T is a forest of order n with maximum degree Δ(T)≥n+23, then h(T)={Δ(T)+2,if T has non-adjacent vertices of degree Δ(T);Δ(T)+1,otherwise.Moreover, the proof yields a polynomial-time algorithm for an optimal harmonious coloring of such a forest.