On the location of chromatic zeros of series-parallel graphs

TL;DR

This paper investigates the distribution of chromatic zeros in series-parallel graphs, showing they are dense in certain regions and disproving a conjecture about zeros with large real parts.

Contribution

It extends understanding of chromatic zeros by identifying their density in the half-plane and disproving a conjecture related to vertex degrees and zero locations.

Findings

Chromatic zeros are dense in the half-plane Re(q)>3/2.

Existence of an open region near (0,32/27) free of zeros.

Counterexample to Sokal's conjecture with high-degree vertices and zeros with large real parts.

Abstract

In this paper we consider the zeros of the chromatic polynomial of series-parallel graphs. Complementing a result of Sokal, showing density outside the disk $|q-1|\leq1$, we show density of these zeros in the half plane $\Re(q)>3/2$ and we show there exists an open region $U$ containing the interval $(0,32/27)$ such that $U\setminus\{1\}$ does not contain zeros of the chromatic polynomial of series-parallel graphs. We also disprove a conjecture of Sokal by showing that for each large enough integer $\Delta$ there exists a series-parallel graph for which all vertices but one have degree at most $\Delta$ and whose chromatic polynomial has a zero with real part exceeding $\Delta$.