Discrete Applied Mathematics 121 (2002) 51–60 An upper bound for the minimum number of queens covering the n × n chessboard A.P. Burger, C.M. Mynhardt * Department of Mathematics, University of South Africa, P.O. Box 392, 0003 Pretoria, South Africa Received 3 June 1999; received in revised form 16 March 2001; accepted 2 April 2001 Abstract We show that the minimum number of queens required to cover the n × n chessboard is at most 8 15 n + O(1). ? 2002 Elsevier Science B.V. All rights reserved. Keywords: Chessboard; Queens graph; Queens domination problem 1. Introduction We generally follow the notation and terminology pertaining to domination of [12]. We repeat the main concepts for chessboards here. The queens graph Q n has the squares of the n × n chessboard as its vertices; two squares are adjacent if they are in the same row, column or diagonal. A queen on square x of Q n covers a square y if x and y are adjacent. A set D of squares is a dominating set of Q n if every square of Q n is either in D or adjacent to a square in D, i.e., if a set of queens, one on each square in D, covers the board. If no two squares of the dominating set D are adjacent, then D is an independent dominating set. The domination number (Q n )(independent domination number i(Q n )) of Q n is the minimum size amongst all dominating (independent dominating) sets of Q n . It is easily seen that (Q n ) 6 i(Q n ) for all n. Chessboard domination problems initiated the study of dominating sets of graphs, at rst rather informally until the topic of domination was given formal mathematical denition with the publication of the books by Berge [2] and Ore [14] in 1962. That even the original chessboard domination problems are astonishingly dicult is apparent in view of the fact that so few of these problems have been solved completely. The unsolved classical problems were important in motivating the revival of the study of * Corresponding author. E-mail address: mynhacm@hotmail.com, mynhacm@unisa.ac.za (C.M. Mynhardt). 0166-218X/02/$ - see front matter ? 2002 Elsevier Science B.V. All rights reserved. PII: S0166-218X(01)00244-X