Harmonious colourings of graphs

Let h(G,λ) denote the number of all λ-harmonious colourings of G. In this paper we analyse the expression h(G,λ) as a function of a variable λ. We observe that this is a polynomial in λ of degree ∣V(G)∣, with a zero constant term. Moreover, we present a reduction formula for calculating h(G,λ). Using reducing steps we show the meaning of some coefficients of h(G,λ) and prove the Nordhaus-Gaddum type theorem, which states that for a graph G with diameter greater than two h(G)+12χ(G2¯)≤∣V(G)∣, whereχ(G2¯) is the chromatic number of the complement of the square of a graph G. Also the notion of harmonious uniqueness is discussed.