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# Galois groups of chromatic polynomials of strongly non-clique-separable graphs of order at most 10

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posted on 2022-07-25, 00:21 authored by K MorganThe chromatic polynomial P(G, λ) gives the number of proper colourings of a graph in at most λ colours. A graph G is clique-separable if it can be obtained by identifying an r-clique in a graph H

_{ 1}with an r-clique in a graph H_{ 2}. In this case the chromatic polynomial of G is P(G, λ) = P(H_{ 1}, λ)P(H_{ 2}, λ)/P(K_{ r}, λ). This paper tabulates the Galois groups of all chromatic polynomials of degree at most 10 (excluding chromatic polynomials of clique-separable graphs). We explicitly list the chromatic polynomials of graphs of order at most 8 with their Galois group. In addition, a summary of the Galois groups of chromatic polynomials of order 5 ≤ n ≤ 10 is given. This includes the number of chromatic polynomials and the number of graphs with each Galois group for a given n.