scholarly journals New bounds for chromatic polynomials and chromatic roots

2015 ◽  
Vol 338 (11) ◽  
pp. 1938-1946 ◽  
Author(s):  
Jason Brown ◽  
Aysel Erey
Keyword(s):  
Chromatic Polynomials ◽  
Chromatic Roots

scholarly journals Density of Chromatic Roots in Minor-Closed Graph Families

2018 ◽  
Vol 27 (6) ◽  
pp. 988-998 ◽  
Author(s):  
THOMAS J. PERRETT ◽  
CARSTEN THOMASSEN
Keyword(s):  
Planar Graph ◽  
Planar Graphs ◽  
Classical Result ◽  
Golden Ratio ◽  
Chromatic Polynomial ◽  
Closed Graph ◽  
Small Interval ◽  
Chromatic Polynomials ◽  
Graph Families ◽  
Chromatic Roots

We prove that the roots of the chromatic polynomials of planar graphs are dense in the interval between 32/27 and 4, except possibly in a small interval around τ + 2 where τ is the golden ratio. This interval arises due to a classical result of Tutte, which states that the chromatic polynomial of every planar graph takes a positive value at τ + 2. Our results lead us to conjecture that τ + 2 is the only such number less than 4.

scholarly journals Chromatic roots as algebraic integers

2012 ◽  
Vol DMTCS Proceedings vol. AR,... (Proceedings) ◽  
Author(s):  
Adam Bohn
Keyword(s):  
Algebraic Integer ◽  
Algebraic Integers ◽  
Chromatic Polynomials ◽  
Integer Multiple ◽  
Chromatic Root ◽  
Il Y A ◽  
International Audience ◽  
The Subject ◽  
Large Families ◽  
Chromatic Roots

International audience A chromatic root is a zero of the chromatic polynomial of a graph. At a Newton Institute workshop on Combinatorics and Statistical Mechanics in 2008, two conjectures were proposed on the subject of which algebraic integers can be chromatic roots, known as the ``$α +n$ conjecture'' and the ``$nα$ conjecture''. These say, respectively, that given any algebraic integer α there is a natural number $n$ such that $α +n$ is a chromatic root, and that any positive integer multiple of a chromatic root is also a chromatic root. By computing the chromatic polynomials of two large families of graphs, we prove the $α +n$ conjecture for quadratic and cubic integers, and show that the set of chromatic roots satisfying the nα conjecture is dense in the complex plane. Une racine chromatique est un zéro du polynôme chromatique d'un graphe. A un atelier au Newton Institute sur la combinatoire et la mécanique statistique en 2008, deux conjectures ont été proposées dont le sujet des entiers algébriques peut être racines chromatiques, connus sous le nom ``la conjecture $α + n$'' et ``la conjecture $n α$ ''. Les conjectures veulent dire, respectivement, que pour chaque entier algébrique $α$ il y a un nombre entier naturel $n$, tel que $α + n$ est une racine chromatique, et que chaque multiple entier positif d'une racine chromatique est aussi une racine chromatique . En calculant les polynômes chromatiques de deux grandes familles de graphes, on prouve la conjecture $α + n$ pour les entiers quadratiques et cubiques, et montre que l'ensemble des racines chromatiques qui confirme la conjecture $nα$ est dense dans le plan complexe.

scholarly journals Chromatic Bounds on Orbital Chromatic Roots

10.37236/4412 ◽  
2014 ◽  
Vol 21 (4) ◽  
Author(s):  
Dae Hyun Kim ◽  
Alexander H. Mun ◽  
Mohamed Omar
Keyword(s):  
Positive Integer ◽  
Real Root ◽  
Chromatic Polynomial ◽  
Outerplanar Graphs ◽  
Chromatic Polynomials ◽  
The Real ◽  
Real Roots ◽  
Chromatic Roots

Given a group $G$ of automorphisms of a graph $\Gamma$, the orbital chromatic polynomial $OP_{\Gamma,G}(x)$ is the polynomial whose value at a positive integer $k$ is the number of orbits of $G$ on proper $k$-colorings of $\Gamma.$ Cameron and Kayibi introduced this polynomial as a means of understanding roots of chromatic polynomials. In this light, they posed a problem asking whether the real roots of the orbital chromatic polynomial of any graph are bounded above by the largest real root of its chromatic polynomial. We resolve this problem in a resounding negative by not only constructing a counterexample, but by providing a process for generating families of counterexamples. We additionally begin the program of finding classes of graphs whose orbital chromatic polynomials have real roots bounded above by the largest real root of their chromatic polynomials; in particular establishing this for many outerplanar graphs.

scholarly journals Approximations for chromatic polynomials

1976 ◽  
Vol 20 (1) ◽  
pp. 5-19 ◽  
Author(s):  
N.L Biggs ◽  
G.H.J Meredith
Keyword(s):  
Chromatic Polynomials

scholarly journals On Weak Chromatic Polynomials of Mixed Graphs

2013 ◽  
Vol 31 (1) ◽  
pp. 91-98 ◽  
Author(s):  
Matthias Beck ◽  
Daniel Blado ◽  
Joseph Crawford ◽  
Taïna Jean-Louis ◽  
Michael Young
Keyword(s):  
Chromatic Polynomials ◽  
Mixed Graphs

scholarly journals Chromatic roots—some observations and conjectures

1980 ◽  
Vol 29 (2) ◽  
pp. 161-167 ◽  
Author(s):  
E.J. Farrell
Keyword(s):  
Chromatic Roots

scholarly journals Note on chromatic polynomials of the threshold graphs

2019 ◽  
Vol 7 (2) ◽  
pp. 217-224
Author(s):  
Noureddine Chikh ◽  
◽  
Miloud Mihoubi ◽  
Keyword(s):  
Chromatic Polynomials ◽  
Threshold Graphs

scholarly journals Algebraic Properties of Chromatic Roots

10.37236/6578 ◽  
2017 ◽  
Vol 24 (1) ◽  
Author(s):  
Peter J. Cameron ◽  
Kerri Morgan
Keyword(s):  
Algebraic Integer ◽  
Tutte Polynomial ◽  
Chromatic Polynomial ◽  
Galois Groups ◽  
Algebraic Numbers ◽  
Isaac Newton ◽  
Isaac Newton Institute ◽  
Chromatic Root ◽  
The Subject ◽  
Chromatic Roots

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 $\alpha$, there is a natural number n such that $\alpha+n$ 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.

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