A majority coloring of a directed graph is a vertex-coloring in which every vertex has the same color as at most half of its out-neighbors. Kreutzer et al. conjectured that every digraph is majority 3-colorable. For an integer k ≥ 2, \({1 \over {k}}\) -majority coloring of a directed graph is a vertex-coloring in which every vertex v has the same color as at most \({1 \over {k}}{d^{+}}(v)\) of its out-neighbors. Girão et al. proved that every digraph admits a \({1 \over {k}}\) -majority 2k-coloring. In this paper, we prove that Kreutzer’s conjecture is true for digraphs under some conditions, which improves Kreutzer’s results, also we obtained some results of \({1 \over {k}}\) -majority coloring of digraphs. Moreover, we discuss the majority 3-coloring of random digraphs with some conditions.
