Galois groups of chromatic polynomials

The chromatic polynomial P(G, λ) gives the number of ways a graph G can be properly coloured in at most λ colours. In this article we give a summary of the Galois groups of all chromatic polynomials of strongly non-clique-separable graphs of order at most 10 and all chromatic polynomial of non-clique-separable θ-graphs of order at most 19. We then consider a number of operations on graphs and show that under some conditions these operations give a graph that has a chromatic polynomial with the same Galois group as the chromatic polynomial of the original graph. Now it is clear that if two polynomials have solvable Galois groups, then the product of these polynomials is also solvable. However, it is not usually the case that the sum of two polynomials with solvable Galois groups has a solvable Galois group. We give cases where P(G', λ) = P(G, λ) + P(Q, λ) where P(G, λ) and P(G', λ) have the same solvable Galois group and P(Q, λ) has the trivial Galois group. Although such a G and G' can easily be found when their Galois group is the trivial group, we give a family of graphs that satisfy this expression where the Galois group of P(G, λ) and P(G', λ) is S2 and another family where the Galois group is D(5). The latter family is particularly interesting as most of the chromatic polynomials of degree at most 10 have symmetric Galois groups and occurrences of nonsymmetric groups are quite rare.