### Abstract

The 3/4--Game Total Domination Conjecture posed by Henning, Klavžar, and Rall [Combinatorica, (2016)] states that if G is a graph on n vertices in which every component contains at least three vertices, then γtg(G)≤3/4n, where γtg(G) denotes the game total domination number of G. Motivated by this conjecture, we raise the problem to a higher level by introducing a transversal game in hypergraphs. We define the game transversal number, φg(H), of a hypergraph H, and prove that if every edge of H has size at least 2, and H ≇C_{4}, then φ_{g}(H) ≥ 4/11 (n_{H} + m_{H}), where n_{H} and m_{H} denote the number of vertices and edges, respectively, in H. Further, we characterize the hypergraphs achieving equality in this bound. As an application of this result, we prove that if G is a graph on n vertices with minimum degree at least 2, then γtg(G) < 8/11 n. As a consequence of this result, the 3/4-Game Total Domination Conjecture is true over the class of graphs with minimum degree at least 2.

Original language | English |
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Pages (from-to) | 1830-1847 |

Number of pages | 18 |

Journal | SIAM Journal on Discrete Mathematics |

Volume | 30 |

Issue number | 3 |

DOIs | |

Publication status | Published - 2016 |

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### Keywords

- Game transversal
- Hypergraph
- Total domination game
- Transversal

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*SIAM Journal on Discrete Mathematics*,

*30*(3), 1830-1847. https://doi.org/10.1137/15M1049361