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 number h ( 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 ) + 1 2 χ ( G 2 ¯ ) ≤ ∣ V ( G ) ∣ , where χ ( G 2 ¯ ) is the chromatic number of the complement of the square of a graph G . Also the notion of harmonious uniqueness is discussed.
Published Version
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