An improved upper bound for queens domination numbers (Q1810636)

The queen's graph \(Q_n\) has the squares of the \(n \times n\) chessboard as its vertices; two vertices are adjacent if the corresponding squares are in the same row, column, or diagonal. The minimum size of a dominating set of \(Q_n\) is denoted by \(\gamma(Q_n)\). The authors show that if, for some fixed \(k\), there is a dominating set of \(Q_{4k+1}\) of a certain type with cardinality \(2k+1\), then for any \(n\) large enough, \(\gamma(Q_n)\leq n(3k+5)/(6k+3)+O(1)\), improving slightly on the results of \textit{W. D. Weakley} [Discrete Math. 242, 229-243 (2002; Zbl 0988.05069)]. A similar result is also obtained for the toroidal queen's graph.