Harmonious colourings of graphs

Abstract

A λ-harmonious colouring of a graph G is a mapping from V(G) into {1,…,λ} that assigns colours to the vertices of G such that each vertex has exactly one colour, adjacent vertices have different colours, and any two edges have different colour pairs. The harmonious chromatic numberh(G) of a graph G is the least positive integer λ, such that there exists a λ-harmonious colouring of G.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.