AI 中文总结
该研究构造了163×163棋盘的显式82个皇后A型1-覆盖，确定γ(Q₁₆₃)=82，并利用其得到更优的皇后图覆盖数渐近系数，还提供了坐标与Python验证器。
AI 中文摘要
皇后图Qₙ以n×n棋盘的方格为顶点，邻接定义为共行、共列或共对角线。我们在Q₁₆₃上给出了一组显式的82个皇后。在中心坐标中，所有皇后坐标均为奇数，每一个奇数行和奇数列各恰好被占据一次，且被占据的行、列和对角线满足Ostergard与Weakley定义的A型1-覆盖条件。直接独立验证检查了全部163²=26569个棋盘方格，未发现任何未被覆盖的方格，因此γ(Q₁₆₃)≤82。此处适用的Finozhenok-Weakley下界γ(Qₙ)≥⌈n/2⌉给出匹配不等式，故γ(Q₁₆₃)=82。对于该覆盖，Ostergard与Weakley定义的参数为e=16、f=15、u=24；完整的差对角线与和对角线多重集在本文中展示。该覆盖还为他们的A型覆盖放大定理提供了显式有限输入，给出γ(Q_N)≤(17/33)N+O(1)。该系数改进了之前的A型系数69/133以及Burger与Mynhardt提出的后续通用系数101/195。本文包含了坐标和一个完整的标准库Python验证器。
英文摘要
The queen's graph $Q_n$ has the squares of the $n\times n$ chessboard as vertices, with adjacency defined by a common row, column, or diagonal. We give an explicit set of $82$ queens on $Q_{163}$. In centered coordinates, all queen coordinates are odd, every odd row and odd column is occupied exactly once, and the occupied rows, columns, and diagonals satisfy the conditions for a type A $1$-cover in the terminology of Ostergard and Weakley. A direct independent verification checks every one of the $163^2=26{,}569$ board squares and finds none uncovered. Hence $γ(Q_{163})\leq82$. The Finozhenok-Weakley lower bound $γ(Q_n)\geq\lceil n/2\rceil$, valid here, gives the matching inequality and therefore $γ(Q_{163})=82$. For this cover, the parameters defined by Ostergard and Weakley are $e=16$, $f=15$, and $u=24$; the complete difference- and sum-diagonal multisets are displayed in the paper. It consequently also supplies an explicit finite input to their amplification theorem for type A covers, giving $γ(Q_N)\leq(17/33)N+O(1)$. This last coefficient improves both the earlier type A coefficient $69/133$ and the subsequent general coefficient $101/195$ of Burger and Mynhardt. The coordinates and a complete standard-library Python verifier are included.
Comments6 pages; includes complete coordinates, a standard-library Python verifier, and a machine-readable certificate. AI-assisted programming and manuscript preparation are disclosed in the paper
