On the location of chromatic zeros of series-parallel graphs

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 $|q-1|\le 1$, 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 $\Delta$.

arXiv commentSmall changes based on referee comments. Accepted for publication in the Electronic Journal of Combinatorics