### Abstract

Let H=(V,E) be a hypergraph with vertex set V and edge set E of order n_{H}=|V| and size m_{H}=|E|. A transversal in H is a subset of vertices in H that has a nonempty intersection with every edge of H. A vertex hits an edge if it belongs to that edge. The transversal game played on H involves of two players, Edge-hitter and Staller, who take turns choosing a vertex from H. Each vertex chosen must hit at least one edge not hit by the vertices previously chosen. The game ends when the set of vertices chosen becomes a transversal in H. Edge-hitter wishes to minimize the number of vertices chosen in the game, while Staller wishes to maximize it. The game transversal number, τ_{g}(H), of H is the number of vertices chosen when Edge-hitter starts the game and both players play optimally. We compare the game transversal number of a hypergraph with its transversal number, and also present an important fact concerning the monotonicity of τ_{g}, that we call the Transversal Continuation Principle. It is known that if H is a hypergraph with all edges of size at least 2, and H is not a 4-cycle, then τ_{g}(H)≤[formula parsented](n_{H}+m_{H}); and if H is a (loopless) graph, then τ_{g}(H)≤[formula parsented](n_{H}+m_{H}+1). We prove that if H is a 3-uniform hypergraph, then τ_{g}(H)≤[formula parsented](n_{H}+m_{H}), and if H is 4-uniform, then τ_{g}(H)≤[formula parsented](n_{H}+m_{H}).

Original language | English |
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Pages (from-to) | 34-50 |

Number of pages | 17 |

Journal | European Journal of Combinatorics |

Volume | 59 |

DOIs | |

Publication status | Published - Jan 1 2017 |

### ASJC Scopus subject areas

- Discrete Mathematics and Combinatorics

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## Cite this

*European Journal of Combinatorics*,

*59*, 34-50. https://doi.org/10.1016/j.ejc.2016.07.003