### 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 = ∪ _{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.

Original language | English |
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Pages (from-to) | 278-292 |

Number of pages | 15 |

Journal | Journal of Graph Theory |

Volume | 71 |

Issue number | 3 |

DOIs | |

Publication status | Published - Nov 2012 |

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### ASJC Scopus subject areas

- Geometry and Topology

### Cite this

*Journal of Graph Theory*,

*71*(3), 278-292. https://doi.org/10.1002/jgt.20646

**Gallai colorings and domination in multipartite digraphs.** / Gyárfás, A.; Simonyi, Gábor; Tóth, Ágnes.

Research output: Contribution to journal › Article

*Journal of Graph Theory*, vol. 71, no. 3, pp. 278-292. https://doi.org/10.1002/jgt.20646

}

TY - JOUR

T1 - Gallai colorings and domination in multipartite digraphs

AU - Gyárfás, A.

AU - Simonyi, Gábor

AU - Tóth, Ágnes

PY - 2012/11

Y1 - 2012/11

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=84865738581&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84865738581&partnerID=8YFLogxK

U2 - 10.1002/jgt.20646

DO - 10.1002/jgt.20646

M3 - Article

AN - SCOPUS:84865738581

VL - 71

SP - 278

EP - 292

JO - Journal of Graph Theory

JF - Journal of Graph Theory

SN - 0364-9024

IS - 3

ER -