### Abstract

An edge coloring of a tournament T with colors 1, 2, ..., k is called k- transitive if the digraph T(i) defined by the edges of color i is transitively oriented for each 1 ≤ i≤ k. We explore a conjecture of the second author: For each positive integer k there exists a (least) p(k) such that every k-transitive tournament has a dominating set of at most p(k) vertices. We show how this conjecture relates to other conjectures and results. For example, it is a special case of a well-known conjecture of Erdos, Sands, Sauer and Woodrow [14] (so the conjecture is interesting even if false). We show that the conjecture implies a stronger conjecture, a possible extension of a result of Bárány and Lehel on covering point sets by boxes. The principle used leads also to an upper bound O(22d-1dlogd) on the d-dimensional box-cover number that is better than all previous bounds, in a sense close to best possible. We also improve the best bound known in 3 dimensions from 3^{14} to 64 and propose possible further improvements through finding the maximum domination number over parity tournaments.

Original language | English |
---|---|

Pages (from-to) | 1-11 |

Number of pages | 11 |

Journal | Journal of Combinatorial Theory. Series B |

Volume | 107 |

Issue number | 1 |

DOIs | |

Publication status | Published - 2014 |

### Fingerprint

### ASJC Scopus subject areas

- Discrete Mathematics and Combinatorics
- Theoretical Computer Science
- Computational Theory and Mathematics

### Cite this

*Journal of Combinatorial Theory. Series B*,

*107*(1), 1-11. https://doi.org/10.1016/j.jctb.2014.02.001

**Domination in transitive colorings of tournaments.** / Pálvölgyi, Dömötör; Gyárfás, A.

Research output: Article

*Journal of Combinatorial Theory. Series B*, vol. 107, no. 1, pp. 1-11. https://doi.org/10.1016/j.jctb.2014.02.001

}

TY - JOUR

T1 - Domination in transitive colorings of tournaments

AU - Pálvölgyi, Dömötör

AU - Gyárfás, A.

PY - 2014

Y1 - 2014

N2 - An edge coloring of a tournament T with colors 1, 2, ..., k is called k- transitive if the digraph T(i) defined by the edges of color i is transitively oriented for each 1 ≤ i≤ k. We explore a conjecture of the second author: For each positive integer k there exists a (least) p(k) such that every k-transitive tournament has a dominating set of at most p(k) vertices. We show how this conjecture relates to other conjectures and results. For example, it is a special case of a well-known conjecture of Erdos, Sands, Sauer and Woodrow [14] (so the conjecture is interesting even if false). We show that the conjecture implies a stronger conjecture, a possible extension of a result of Bárány and Lehel on covering point sets by boxes. The principle used leads also to an upper bound O(22d-1dlogd) on the d-dimensional box-cover number that is better than all previous bounds, in a sense close to best possible. We also improve the best bound known in 3 dimensions from 314 to 64 and propose possible further improvements through finding the maximum domination number over parity tournaments.

AB - An edge coloring of a tournament T with colors 1, 2, ..., k is called k- transitive if the digraph T(i) defined by the edges of color i is transitively oriented for each 1 ≤ i≤ k. We explore a conjecture of the second author: For each positive integer k there exists a (least) p(k) such that every k-transitive tournament has a dominating set of at most p(k) vertices. We show how this conjecture relates to other conjectures and results. For example, it is a special case of a well-known conjecture of Erdos, Sands, Sauer and Woodrow [14] (so the conjecture is interesting even if false). We show that the conjecture implies a stronger conjecture, a possible extension of a result of Bárány and Lehel on covering point sets by boxes. The principle used leads also to an upper bound O(22d-1dlogd) on the d-dimensional box-cover number that is better than all previous bounds, in a sense close to best possible. We also improve the best bound known in 3 dimensions from 314 to 64 and propose possible further improvements through finding the maximum domination number over parity tournaments.

KW - Domination

KW - Transitive tournaments

UR - http://www.scopus.com/inward/record.url?scp=84901822848&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84901822848&partnerID=8YFLogxK

U2 - 10.1016/j.jctb.2014.02.001

DO - 10.1016/j.jctb.2014.02.001

M3 - Article

AN - SCOPUS:84901822848

VL - 107

SP - 1

EP - 11

JO - Journal of Combinatorial Theory. Series B

JF - Journal of Combinatorial Theory. Series B

SN - 0095-8956

IS - 1

ER -