Chromatic Factors
نویسندگان
چکیده
The chromatic polynomial P (G, λ) gives the number of proper colourings of a graph G in at most λ colours. If P (G, λ) = P (H1, λ)P (H2, λ) /P (Kr, λ), then G is said to have a chromatic factorisation of order r with chromatic factors H1 and H2. It is clear that, if 0 ≤ r ≤ 2, any H1 6∼= Kr with chromatic number χ(H1) ≥ r is the chromatic factor of some chromatic factorisation of order r. We show that every H1 6∼= K3 with χ(H1) ≥ 3, even when H1 contains no triangles, is the chromatic factor of some chromatic factorisation of order 3 and give a certificate of factorisation for this chromatic factorisation. This certificate shows in a sequence of six steps using some basic properties of chromatic polynomials that a graph G has a chromatic factorisation with one of the chromatic factors being H1. This certificate is one of the shortest known certificates of factorisation, excluding the trivial certificate for chromatic factorisations of clique-separable graphs.
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تاریخ انتشار 2009