  For a graph $G=(V,E)$, $k\in \mathbb{N}$, and a complex number $w$ the
partition function of the univariate Potts model is defined as \[ {\bf
Z}(G;k,w):=\sum_{\phi:V\to [k]}\prod_{\substack{uv\in E \\ \phi(u)=\phi(v)}}w,
\] where $[k]:=\{1,\ldots,k\}$. In this paper we give zero-free regions for the
partition function of the anti-ferromagnetic Potts model on bounded degree
graphs. In particular we show that for any $\Delta\in \mathbb{N}$ and any
$k\geq e\Delta+1$, there exists an open set $U$ in the complex plane that
contains the interval $[0,1)$ such that ${\bf Z}(G;k,w)\neq 0$ for any $w\in U$
and any graph $G$ of maximum degree at most $\Delta$. (Here $e$ denotes the
base of the natural logarithm.) For small values of $\Delta$ we are able to
give better results.
  As an application of our results we obtain improved bounds on $k$ for the
existence of deterministic approximation algorithms for counting the number of
proper $k$-colourings of graphs of small maximum degree.
