The harmonious chromatic number of complete r-ary trees

Abstract

A harmonious colouring of a simple graph 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. We define Q( m) to be the least positive integer k such that k 2 ⩾m . Then h( G)⩾ Q( m) for any graph G with m edges. We consider the complete r-ary tree of height H, denoted T r, H . We show that for any r⩾2, H⩾3 , if m is the number of edges of T r, H , then h( T r, H )= Q( m), except that h( T 2,3)=7.