Gallai colorings and domination in multipartite digraphs

TL;DR

This paper investigates the structure of Gallai colorings and domination in multipartite digraphs, establishing bounds on dominating sets based on independence properties, with applications to edge colorings avoiding 3-colored triangles.

Contribution

It proves the existence of a bound on dominating set size in certain digraphs related to Gallai colorings, linking independence and domination properties.

Findings

Existence of a function h(beta(D)) bounding dominating set size

Application to generalized Gallai colorings and triangle-free edge colorings

Resolution of a problem related to 3-colored triangle avoidance

Abstract

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.