A note on the real part of complex chromatic roots

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 nn vertices is n−1n−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−1n−1 and the tree-width of the graph.In this article we show that for all k≥2k≥2 there exist infinitely many graphs GG with tree-width kk such that GG has non-real chromatic roots zz with ℜ(z)>kℜ(z)>k. We also discuss the weaker conjecture and prove it for graphs GG with χ(G)≥n−3χ(G)≥n−3.