Constructing All Birthday 3 Games as Digraphs
Fuente:
arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866908356560551936 |
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| author | Clow, Alexander Davies, Alfie McKay, Neil Anderson |
| author_facet | Clow, Alexander Davies, Alfie McKay, Neil Anderson |
| contents | Recently, Clow and McKay proved that the Digraph Placement ruleset is universal for normal play: for all normal play combinatorial games $X$, there is a Digraph Placement game $G$ with $G=X$. Clow and McKay also showed that the 22 game values born by day 2 correspond to Digraph Placement games with at most 4 vertices. This bound is best possible. We extend this work using a combination of exhaustive and random searches to demonstrate all 1474 values born by day 3 correspond to Digraph Placement games on at most 8 vertices. We provide a combinatorial proof that this bound is best possible. We conclude by giving improved bounds on the number of vertices required to construct all game values born by days 4 and 5. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_06206 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Constructing All Birthday 3 Games as Digraphs Clow, Alexander Davies, Alfie McKay, Neil Anderson Combinatorics 91A46 (Primary) 05C20 (Secondary) Recently, Clow and McKay proved that the Digraph Placement ruleset is universal for normal play: for all normal play combinatorial games $X$, there is a Digraph Placement game $G$ with $G=X$. Clow and McKay also showed that the 22 game values born by day 2 correspond to Digraph Placement games with at most 4 vertices. This bound is best possible. We extend this work using a combination of exhaustive and random searches to demonstrate all 1474 values born by day 3 correspond to Digraph Placement games on at most 8 vertices. We provide a combinatorial proof that this bound is best possible. We conclude by giving improved bounds on the number of vertices required to construct all game values born by days 4 and 5. |
| title | Constructing All Birthday 3 Games as Digraphs |
| topic | Combinatorics 91A46 (Primary) 05C20 (Secondary) |
| url | https://arxiv.org/abs/2505.06206 |