p-Adic Roots of Chromatic Polynomials

Abstract

The complex roots of the chromatic polynomial $P_{G}(x)$ of a graph G have been well studied, but the p-adic roots have received no attention as yet. We consider these roots, specifically the roots in the ring $\mathbb{Z}_p$ of p-adic integers. We first describe how the existence of p-adic roots is related to the p-divisibility of the number of colourings of a graph—colourings by at most k colours and also ones by exactly k colours. Then we turn to the question of the circumstances under which $P_{G}(x)$ splits completely over $\mathbb{Z}_p$, giving some generalities before considering in detail an infinite family of graphs whose chromatic polynomials have been discovered, by Morgan (LMS J Comput Math 15, 281–307, 2012), to each have a cubic abelian splitting field.