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Strengthened Brooks' Theorem for Digraphs of Girth at least Three
2011
Electronic Journal of Combinatorics
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})
doi:10.37236/682
fatcat:oiusc4lzibbvlomjcwxpusg2rq