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 () = 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. We show that for any r = 2, H = 3, if m is the number of edges of T, then h(T) = Q(m), except that h(T) = 7.
|Number of pages||17|
|Publication status||Published - 1999|