The Harmonious Chromatic Number and the Achromatic Number

Abstract

Summary The harmonious chromatic number of a graph is the least number of colours in a vertex colouring such that each pair of colours appears on at most one edge. The achromatic number of a graph is the greatest number of colours in a vertex colouring such that each pair of colours appears on at least one edge. This paper is a survey of what is known about these two parameters, in particular we look at upper and lower bounds, special classes of graphs and complexity issues. Introduction A short survey of harmonious colourings was given by Wilson [80] in 1990. Since then a number of new results have appeared, and the close relationship between harmonious chromatic number and achromatic number has been observed. The purpose of this new survey is to outline what is known about these parameters, and suggest some open problems. A more detailed summary of results on the achromatic number, with a rather different emphasis, can be found in the forthcoming survey by Hughes and MacGillivray [51]. We begin with the definitions of the two parameters. Definitions A harmonious colouring of a graph G is a proper vertex colouring of G such that, for any pair of colours, there is at most one edge of G whose endpoints are coloured with this pair of colours. The harmonious chromatic number of G, denoted h(G), is the least number of colours in a harmonious colouring of G.