New upper bounds on harmonious colourings

We present an improved upper bound on the harmonious chromatic number of an arbitrary graph. We also consider “Fragmentable” classes of graphs (an example is the class of planar graphs) which are, roughly speaking, graphs which can be decomposed into bounded sized components by removing a small proportion of the vertices. We show that for such graphs of bounded degree the harmonious chromatic number is close to the lower bound (2m)^1/2 , where m is the number of edges.