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Chromatic Bounds on Orbital Chromatic Roots

Kim, Dae Hyun and Mun, Alexander H. and Omar, Mohamed (2014) Chromatic Bounds on Orbital Chromatic Roots. Electronic Journal of Combinatorics, 21 (4). Art. No. P4.17. ISSN 1077-8926. http://resolver.caltech.edu/CaltechAUTHORS:20141201-101314782

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Abstract

Given a group G of automorphisms of a graph Γ, the orbital chromatic polynomial OPΓ,G(x) is the polynomial whose value at a positive integer k is the number of orbits of G on proper k-colorings of Γ. 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.


Item Type:Article
Related URLs:
URLURL TypeDescription
http://www.combinatorics.org/ojs/index.php/eljc/article/view/v21i4p17PublisherArticle
Additional Information:© 2014 The Authors. Submitted: May 30, 2014; Accepted: Oct 10, 2014; Published: Oct 23, 2014. The authors express sincere thanks to the anonymous referees for providing a number of helpful suggestions to improve the presentation of this paper. Supported by Summer Undergraduate Research Fellowships at the California Institute of Technology.
Funders:
Funding AgencyGrant Number
Summer Undergraduate Research Fellowship (SURF), CaltechUNSPECIFIED
Classification Code:Mathematics Subject Classiffications: 05C15, 05C30, 05C31
Record Number:CaltechAUTHORS:20141201-101314782
Persistent URL:http://resolver.caltech.edu/CaltechAUTHORS:20141201-101314782
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:52209
Collection:CaltechAUTHORS
Deposited By: Jason Perez
Deposited On:02 Dec 2014 00:26
Last Modified:02 Dec 2014 00:26

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