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{\bf Ararat Harutyunyan and  Bojan Mohar}
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{\bf Strengthened Brooks' Theorem for Digraphs of Girth at least Three}
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Brooks' Theorem states that a connected graph $G$ of maximum
degree $\Delta$ has chromatic number at most $\Delta$, unless $G$
is an odd cycle or a complete graph. A result of Johansson shows
that if $G$ is triangle-free, then the chromatic number drops to
$O(\Delta / \log \Delta)$. In this paper, we derive a weak analog
for the chromatic number of digraphs. We show that every
(loopless) digraph $D$ without directed cycles of length two has
chromatic number $\chi(D) \leq (1-e^{-13}) \tilde{\Delta}$, where
$\tilde{\Delta}$ is the maximum geometric mean of the out-degree and
in-degree of a vertex in $D$, when $\tilde{\Delta}$ is sufficiently
large. As a corollary it is proved that there exists an absolute
constant $\alpha < 1$ such that $\chi(D) \leq \alpha (\tilde{\Delta} +
1)$ for every $\tilde{\Delta} > 2$.

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