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Published October 23, 2014 | Published
Journal Article Open

Chromatic Bounds on Orbital Chromatic Roots

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.

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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