  Assume that D is a digraph without cyclic triangles and its vertices are
partitioned into classes A_1,...,A_t of independent vertices. A set
$U=\cup_{i\in S} A_i$ is called a dominating set of size |S| if for any vertex
$v\in \cup_{i\notin S} A_i$ there is a w in U such that (w,v) is in E(D). Let
beta(D) be the cardinality of the largest independent set of D whose vertices
are from different partite classes of D. Our main result says that there exists
a h=h(beta(D)) such that D has a dominating set of size at most h. This result
is applied to settle a problem related to generalized Gallai colorings, edge
colorings of graphs without 3-colored triangles.
