Orthodox queen domination: finite constructions and an asymptotic density gap
Abstract
An orthodox dominating set on an n× n chessboard occupies every row of one parity and every column of a possibly different parity. For n≥400001, 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 Q221 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.
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BibTeX
@misc{kong2026orthodox,
title = {{Orthodox queen domination: finite constructions and an asymptotic density gap}},
author = {Yixiang Kong},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.01513},
eprint = {2609.01513},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.01513},
}
RIS
TY - UNPB
TI - Orthodox queen domination: finite constructions and an asymptotic density gap
AU - Yixiang Kong
PY - 2026
DA - 2026/09/15
PB - arXiv
DO - 10.48550/arxiv.2609.01513
UR - https://arxiv.org/abs/2609.01513
AB - An orthodox dominating set on an n× n chessboard occupies every row of one parity and every column of a possibly different parity. For n≥400001, 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.
ER -