On Harmonious Colouring of Trees

Let G be a simple graph and (G) denote the maximum degree of G. A harmonious colouring of G is a proper vertex colouring such that each pair of colours appears together on at most one edge. The harmonious chromatic number h(G) is the least number of colours in such a colouring. In this paper it is shown that if T is a tree of order n and (T ) n2 , then there exists a harmonious colouring of T with (T ) + 1 colours such that every colour is used at most twice. Thus h(T ) = (T )+1. Moreover, we prove that if T is a tree of order n and (T ) dn2 e, then there exists a harmonious colouring of T with dn2 e+ 1 colours such that every colour is used at most twice. Thus h(T ) dn2 e+ 1.