Year: 2026
Type: preprint
Abstract: 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 $1/2+1/16000$.
Source: arXiv (Cornell University)
Author: Kong, Yixiang
Institution: Northeastern University
Language: English
Subfield: Discrete Mathematics and Combinatorics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7206197451