A new lower bound on upper irredundance in the queens' graph

The queens’ graph Qn has the squares of the n × n chessboard as its vertices, with two squares adjacent if they are in the same row, column, or diagonal. An irredundant set of queens has the property that each queen in the set attacks at least one square which is attacked by no other queen. IR(Qn) is the cardinality of the largest irredundant set of vertices in Qn. Currently the best lower bound for IR(Qn) is IR(Qn)¿ 2:5n − O(1), while the best upper bound is IR(Qn)6 6n + 6 − 8 √ n+ √ n+ 1 for n¿ 6. Here the lower bound is improved to IR(Qn)¿ 6n − O(n). In particular, it is shown for even k¿ 6 that IR(Qk3 )¿ 6k − 29k − O(k). c © 2002 Elsevier Science B.V. All rights reserved.