Abstract
AbstractThe chromatic polynomialP(G,λ) gives the number of ways a graphGcan be properly coloured in at mostλcolours. This polynomial has been extensively studied in both combinatorics and statistical physics, but there has been little work on its algebraic properties. This paper reports a systematic study of the Galois groups of chromatic polynomials. 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 polynomials of non-clique-separableθ-graphs of order at most 19. Most of these chromatic polynomials have symmetric Galois groups. We give five infinite families of graphs: one of these families has chromatic polynomials with a dihedral Galois group and two of these families have chromatic polynomials with cyclic Galois groups. This includes the first known infinite family of graphs that have chromatic polynomials with the cyclic Galois group of order 3.
Highlights
The chromatic polynomial P (G, λ) gives the number of proper colourings of a graph G in at most λ colours. It was first introduced by Birkhoff [7] in an attempt to prove algebraically the four-colour theorem, that is, that every planar graph is four colourable. This attempt was unsuccessful, the chromatic polynomial has been extensively studied both in graph theory [17, 45] and in statistical mechanics where the Potts model partition function generalises this polynomial
In this paper we look at the Galois groups of chromatic polynomials
In this paper we presented the first results about Galois groups of chromatic polynomials
Summary
The chromatic polynomial P (G, λ) gives the number of proper colourings of a graph G in at most λ colours. The first contribution is to give a summary of our computations of Galois groups of the irreducible non-linear factors of chromatic polynomials of all strongly non-clique-separable graphs of order at most 10 and of θ-graphs of order at most 19. As most chromatic polynomials of strongly non-clique-separable graphs of order at most 10 have symmetric Galois groups, it is interesting to consider cases where the Galois group is not the symmetric Galois group, which leads to our second contribution. The second contribution of this paper is to give some new infinite families of Galois equivalent graphs Three of these families have graphs with chromatic polynomials that have nonsymmetric Galois groups, namely, the cyclic groups, C(3) ∼= A3 and C(4), of orders 3 and 4 respectively, and the dihedral group D(4). Each family is shown to have Galois group S1, S2, C(3) ∼= A3, C(4) and D(4) respectively
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