Non-bipartite chromatic factors

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 ≅ Kr with chromatic number χ( H1 ) < r is the chromatic factor of some chromatic factorisation of order r. We show that every H1 ≅ 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 seven 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.