TY - JOUR
AU - Buckingham, Paul
AB - The complex roots of the chromatic polynomial PG(x)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$P_{G}(x)$$\end{document} 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 Zp\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\mathbb{Z}_p$$\end{document} 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 PG(x)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$P_{G}(x)$$\end{document} splits completely over Zp\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\mathbb{Z}_p$$\end{document}, 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.
TI - p-Adic Roots of Chromatic Polynomials
JF - Graphs and Combinatorics
DO - 10.1007/s00373-020-02171-y
DA - 2020-04-27
UR - https://www.deepdyve.com/lp/springer-journals/p-adic-roots-of-chromatic-polynomials-DgToFaRmW1
SP - 1111
EP - 1130
VL - 36
IS - 4
DP - DeepDyve
ER -