A note on the real part of complex chromatic roots

TL;DR

This paper investigates the real parts of chromatic roots of graphs, disproving a conjecture that they are bounded above by the tree-width, by constructing graphs with roots exceeding this bound.

Contribution

It demonstrates the existence of graphs with tree-width k whose chromatic roots have real parts greater than k, challenging previous conjectures.

Findings

Existence of graphs with non-real chromatic roots with real parts exceeding tree-width.

Disproof of the conjecture that real parts of chromatic roots are bounded by tree-width.

Proved a weaker conjecture for graphs with chromatic number at least n-3.

Abstract

A {\em 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 {\em 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 [8] 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 $k\geq 2$ there exist infinitely many graphs $G$ with tree-width $k$ such that $G$ has non-real chromatic roots $z$ with $\Re(z)>k$. We also discuss the weaker conjecture and prove it for graphs $G$ with $\chi(G)\geq n-3$.