A Constructive Winning Maker Strategy in the Maker-Breaker $C_4$-Game
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866917705172385792 |
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| author | Sowa, Matthias Srivastav, Anand |
| author_facet | Sowa, Matthias Srivastav, Anand |
| contents | Maker-Breaker subgraph games are among the most famous combinatorial games. For given $n,q \in \mathbb{N}$ and a subgraph $C$ of the complete graph $K_n$, the two players, called Maker and Breaker, alternately claim edges of $K_n$. In each round of the game Maker claims one edge and Breaker is allowed to claim up to $q$ edges. If Maker is able to claim all edges of a copy of $C$, he wins the game. Otherwise Breaker wins. In this work we introduce the first constructive strategy for Maker for the $C_4$-Maker-Breaker game and show that he can win the game if $q < 0.16 n^{2/3}$. According to the theorem of Bednarska and Luczak (2000) $n^{2/3}$ is asymptotically optimal for this game, but the constant given there for a random Maker strategy is magnitudes apart from our constant 0.16. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_04462 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A Constructive Winning Maker Strategy in the Maker-Breaker $C_4$-Game Sowa, Matthias Srivastav, Anand Combinatorics Maker-Breaker subgraph games are among the most famous combinatorial games. For given $n,q \in \mathbb{N}$ and a subgraph $C$ of the complete graph $K_n$, the two players, called Maker and Breaker, alternately claim edges of $K_n$. In each round of the game Maker claims one edge and Breaker is allowed to claim up to $q$ edges. If Maker is able to claim all edges of a copy of $C$, he wins the game. Otherwise Breaker wins. In this work we introduce the first constructive strategy for Maker for the $C_4$-Maker-Breaker game and show that he can win the game if $q < 0.16 n^{2/3}$. According to the theorem of Bednarska and Luczak (2000) $n^{2/3}$ is asymptotically optimal for this game, but the constant given there for a random Maker strategy is magnitudes apart from our constant 0.16. |
| title | A Constructive Winning Maker Strategy in the Maker-Breaker $C_4$-Game |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2405.04462 |