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 ot colours in such a colouring. Let d be a fixed positive integer, and e > 0. We show that there is a natural number M such that if G is any graph with m = M edges and maximum degree at most d, then the harmonious chromatic number h(G) satisfies (2m) = h(G) = (1+e) (2m).
|Number of pages||13|
|Journal||Journal of the London Mathematical Society|
|Publication status||Published - 1997|