Maker playing against an invisible Breaker
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arXiv
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| Main Authors: | , , , , |
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866908472868601856 |
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| author | Clemens, Dennis Hamann, Fabian Mikalački, Mirjana Mogge, Yannick Stojaković, Miloš |
| author_facet | Clemens, Dennis Hamann, Fabian Mikalački, Mirjana Mogge, Yannick Stojaković, Miloš |
| contents | We initiate the study of the phantom version of Maker-Breaker positional games. In a phantom game, the moves of one of the players are hidden from the other player, who still has the complete information. We look at the biased $(a:b)$ Maker-PhantomBreaker games where the board is the edge set of the complete graph on $n$ vertices, $K_n$, and Maker has no information about PhantomBreaker's choices of edges. We give randomized strategies for both players in four classical games: connectivity game, perfect matching game, mindegree-$k$ game and Hamiltonicity game. In particular, we focus on characterizing all biases $(a:b)$ for which Maker wins asymptotically almost surely. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_22519 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Maker playing against an invisible Breaker Clemens, Dennis Hamann, Fabian Mikalački, Mirjana Mogge, Yannick Stojaković, Miloš Combinatorics 05C57, 05C40, 05C45 We initiate the study of the phantom version of Maker-Breaker positional games. In a phantom game, the moves of one of the players are hidden from the other player, who still has the complete information. We look at the biased $(a:b)$ Maker-PhantomBreaker games where the board is the edge set of the complete graph on $n$ vertices, $K_n$, and Maker has no information about PhantomBreaker's choices of edges. We give randomized strategies for both players in four classical games: connectivity game, perfect matching game, mindegree-$k$ game and Hamiltonicity game. In particular, we focus on characterizing all biases $(a:b)$ for which Maker wins asymptotically almost surely. |
| title | Maker playing against an invisible Breaker |
| topic | Combinatorics 05C57, 05C40, 05C45 |
| url | https://arxiv.org/abs/2507.22519 |