Harmonious coloring of trees with large maximum degree

Abstract

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 Delta(T) >= n+2/3, then h(T) = {Delta(T) + 2, if T has non-adjacent vertices of degree Delta(T); Delta(T) + 1, otherwise. Moreover, the proof yields a polynomial-time algorithm for an optimal harmonious coloring of such a forest. (C) 2012 Elsevier B.V. All rights reserved.