### Abstract

A chromatic root is a root of the chromatic polynomial of a graph. Any chromatic root is an algebraic integer. Much is known about the location of chromatic roots in the real and complex numbers, but rather less about their properties as algebraic numbers. This question was the subject of a seminar at the Isaac Newton Institute in late 2008. The purpose of this paper is to report on the seminar and subsequent developments. We conjecture that, for every algebraic integer α, there is a natural number n such that α+n is a chromatic root. This is proved for quadratic integers; an extension to cubic integers has been found by Adam Bohn. The idea is to consider certain special classes of graphs for which the chromatic polynomial is a product of linear factors and one “interesting” factor of larger degree. We also report computational results on the Galois groups of irreducible factors of the chromatic polynomial for some special graphs. Finally, extensions to the Tutte polynomial are mentioned briefly.

Original language | English |
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Article number | #P1.21 |

Number of pages | 14 |

Journal | Electronic Journal of Combinatorics |

Volume | 24 |

Issue number | 1 |

Publication status | Published - 3 Feb 2017 |

### Cite this

*Electronic Journal of Combinatorics*,

*24*(1), [#P1.21].

}

*Electronic Journal of Combinatorics*, vol. 24, no. 1, #P1.21.

**Algebraic properties of chromatic roots.** / Cameron, Peter J.; Morgan, Kerri.

Research output: Contribution to journal › Article › Research › peer-review

TY - JOUR

T1 - Algebraic properties of chromatic roots

AU - Cameron, Peter J.

AU - Morgan, Kerri

PY - 2017/2/3

Y1 - 2017/2/3

N2 - A chromatic root is a root of the chromatic polynomial of a graph. Any chromatic root is an algebraic integer. Much is known about the location of chromatic roots in the real and complex numbers, but rather less about their properties as algebraic numbers. This question was the subject of a seminar at the Isaac Newton Institute in late 2008. The purpose of this paper is to report on the seminar and subsequent developments. We conjecture that, for every algebraic integer α, there is a natural number n such that α+n is a chromatic root. This is proved for quadratic integers; an extension to cubic integers has been found by Adam Bohn. The idea is to consider certain special classes of graphs for which the chromatic polynomial is a product of linear factors and one “interesting” factor of larger degree. We also report computational results on the Galois groups of irreducible factors of the chromatic polynomial for some special graphs. Finally, extensions to the Tutte polynomial are mentioned briefly.

AB - A chromatic root is a root of the chromatic polynomial of a graph. Any chromatic root is an algebraic integer. Much is known about the location of chromatic roots in the real and complex numbers, but rather less about their properties as algebraic numbers. This question was the subject of a seminar at the Isaac Newton Institute in late 2008. The purpose of this paper is to report on the seminar and subsequent developments. We conjecture that, for every algebraic integer α, there is a natural number n such that α+n is a chromatic root. This is proved for quadratic integers; an extension to cubic integers has been found by Adam Bohn. The idea is to consider certain special classes of graphs for which the chromatic polynomial is a product of linear factors and one “interesting” factor of larger degree. We also report computational results on the Galois groups of irreducible factors of the chromatic polynomial for some special graphs. Finally, extensions to the Tutte polynomial are mentioned briefly.

UR - http://www.scopus.com/inward/record.url?scp=85011584272&partnerID=8YFLogxK

M3 - Article

VL - 24

JO - The Electronic Journal of Combinatorics

JF - The Electronic Journal of Combinatorics

SN - 1077-8926

IS - 1

M1 - #P1.21

ER -