A note on the real part of complex chromatic roots

Abstract

A chromatic root is a root of the chromatic polynomial of a graph. While the real chromatic roots have been extensively studied and well understood, little is known about the real parts of chromatic roots. It is not difficult to see that the largest real chromatic root of a graph with n vertices is n−1, and indeed, it is known that the largest real chromatic root of a graph is at most the tree-width of the graph. Analogous to these facts, it was conjectured in Dong et al. (2005) that the real parts of chromatic roots are also bounded above by both n−1 and the tree-width of the graph.In this article we show that for all k≥2 there exist infinitely many graphs G with tree-width k such that G has non-real chromatic roots z with ℜ(z)>k. We also discuss the weaker conjecture and prove it for graphs G with χ(G)≥n−3.