On Chromatic Roots with Negative Real Part

Abstract

A chromatic root is a root of the chromatic polynomial of some graph \(G\). E. Farrell conjectured in \(1980\) that no chromatic root can lie in the left-half plane, and in \(1991\) Read and Royle showed by direct computation that the chromatic polynomials of some graphs do have a root there. These examples, though, yield only finitely many such chromatic roots. Subsequent results by Shrock and Tsang show the existence of chromatic roots of arbitrarily large negative real part. We show that theta graphs with equal path lengths of size at least \(8\) have chromatic roots with negative real part.