Gallai colorings and domination in multipartite digraphs

Assume that D is a digraph without cyclic triangles and its vertices are partitioned into classes A1, . . . , At of independent vertices. A set U = ∪i∈SAi is called a dominating set of size |S| if for any vertex v ∈ ∪i/ ∈SAi there is a w ∈ U such that (w, v) ∈ E(D). Let β(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(β(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. ∗Research partially supported by the Hungarian Foundation for Scientific Research Grant (OTKA) No. K68322. †Research partially supported by the Hungarian Foundation for Scientific Research Grant (OTKA) Nos. K76088 and NK78439. ‡Research partially supported by the Hungarian Foundation for Scientific Research Grant and by the National Office for Research and Technology (Grant number OTKA 67651).