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

  • Authors:

  • Matthew D. Kearse;Peter B. Gibbons

  • Affiliations:

  • Department of Computer Science, University of Auckland, Private Bag 92019, Auckland, New Zealand;Department of Computer Science, University of Auckland, Private Bag 92019, Auckland, New Zealand

  • Venue:
  • Discrete Mathematics
  • Year:
  • 2002

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Abstract

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) ≤ ⌊6n+6 - 8 √n+√n+1⌋ for n ≥ 6. Here the lower bound is improved to IR(Qn) ≥ 6n - O(n2/3). In particular, it is shown for even k ≥ 6 that IR(Qk3)≥ 6k3 - 29k2 - O(k).