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 = ∪ i∈S A i is called a dominating set of size |S| if for any vertex V ∈ ∪ i∉S A i 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.
ASJC Scopus subject areas
- Geometry and Topology