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 + 2 3 , 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.