Q221的无边界独立A型覆盖及皇后支配的改进渐近界
Orthodox queen domination: finite constructions and an asymptotic density gap
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中文总结 AI 辅助
本文构造了Q221的111个无外边缘的互不攻击皇后的A型1-覆盖，确定了皇后支配数和独立支配数均为111，改进了皇后支配的渐近界，并给出了相关的约简与验证方法。
中文摘要 AI 辅助
皇后图$Q_n$以$n\times n$棋盘的方格为顶点，相邻定义为共享同一行、列或对角线。我们给出$Q_{221}$上111个互不攻击的皇后的显式集合，该集合不含位于外排或外列的皇后，且满足Ostergard和Weakley提出的参数为$(e,f,u)=(24,23,31)$的原始A型1-覆盖条件。直接枚举检查了全部$221^2=48841$个棋盘方格，未发现未被覆盖的方格，因此Finozhenok和Weakley的下界给出$\gamma(Q_{221})=i(Q_{221})=111$，Neuhaus此前已证实普通支配下的该等式。A型放大定理的无边界方格分支给出$\gamma(Q_N)\leq(112/221)N+O(1)$和$i(Q_N)\leq(113/221)N+O(1)$，这些系数分别改进了Neuhaus给出的$30/59$和$91/177$。我们还描述了用于获取证书的精确四族匹配模型，赋值对偶恒等式给出无损耗的降成本删除规则，交替允许边测试给出第二种无损耗约简，手稿附带完整坐标、两个独立实现的标准库证书验证器及确定性约简审计。
英文摘要
An orthodox dominating set on an $n\times n$ chessboard occupies every row of one parity and every column of a possibly different parity. For $n\ge400001$, every such set contains more than $(1/2+1/80000)n-2$ queens, on odd and even boards and with attacking or boundary queens allowed. Second-moment estimates and an exact rational line-weight certificate give a stronger bound for $p$-covers; finite diagonal completion and board extension transfer it to orthodox covers. The cost of extending an arbitrary dominating set to an orthodox cover yields an inequality with an explicit defect term. The previously constructed independent, border-free Type-A $1$-cover of $Q_{221}$ with $111$ queens supplies a finite seed. Classical amplification gives ordinary and independent domination upper bounds with coefficients $112/221$ and $113/221$, respectively. Its order 221 is below the density threshold 400001. For admissible seeds whose orders tend to infinity, the lower limit of these coefficients is at least $1/2+1/16000$.
发表机构
- Sydney Smart Technology College, Northeastern University（东北大学悉尼智能科技学院）
机构由 AI 辅助整理，请以论文原文为准。
