On the Structure of 3D Queen Domination
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2026
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| _version_ | 1866908937084731392 |
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| author | Ramani, Mahesh |
| author_facet | Ramani, Mahesh |
| contents | We study the domination number $γ(Q_n^3)$ of the three-dimensional $n \times n \times n$ queen graph. The main result is a stratified theorem computing, for each position type -- corner, edge, face, or interior -- the number of inner-core vertices dominated by a queen, and showing in particular that interior placements dominate strictly more core cells than boundary placements. This yields a symmetry-reduction principle via the octahedral group and complements the standard counting lower bound and layered upper bound, giving $γ(Q_n^3) = Θ(n^2)$. We also certify exact values for $n \leq 6$ via integer linear programming and independent verification. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_03793 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On the Structure of 3D Queen Domination Ramani, Mahesh Combinatorics Discrete Mathematics 05C69, 00A08 G.2.2 We study the domination number $γ(Q_n^3)$ of the three-dimensional $n \times n \times n$ queen graph. The main result is a stratified theorem computing, for each position type -- corner, edge, face, or interior -- the number of inner-core vertices dominated by a queen, and showing in particular that interior placements dominate strictly more core cells than boundary placements. This yields a symmetry-reduction principle via the octahedral group and complements the standard counting lower bound and layered upper bound, giving $γ(Q_n^3) = Θ(n^2)$. We also certify exact values for $n \leq 6$ via integer linear programming and independent verification. |
| title | On the Structure of 3D Queen Domination |
| topic | Combinatorics Discrete Mathematics 05C69, 00A08 G.2.2 |
| url | https://arxiv.org/abs/2604.03793 |