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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.

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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.

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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.
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Summer Undergraduate Research Fellowship (SURF), CaltechUNSPECIFIED
Issue or Number:4
Classification Code:Mathematics Subject Classiffications: 05C15, 05C30, 05C31
Record Number:CaltechAUTHORS:20141201-101314782
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Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:52209
Deposited By: Jason Perez
Deposited On:02 Dec 2014 00:26
Last Modified:03 Oct 2019 07:40

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