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.

对图G的色多项式\（P_ {G}（x）\）的复根进行了很好的研究，但p -adic根至今尚未引起关注。我们考虑这些根，特别是p -adic整数的环\（\ mathbb {Z} _p \）中的根。我们首先介绍如何存在p进制根是关系到p至多一个图的着色的色素数量-divisibility ķ的色彩和那些也恰好ķ颜色。然后我们来讨论\（P_ {G}（x）\）在\（\ mathbb {Z} _p \）上完全分割的情况下的问题。，先给出一些通用性，然后再详细考虑由Morgan（LMS J Comput Math 15，15，281-307，2012）发现的其色多项式的无穷多个图，每个图都有立方阿贝尔分裂场。