OFFSET
1,8
COMMENTS
The queens graph Q(n X m) has the squares of the n X m chessboard as its vertices; two squares are adjacent if they are both in the same row, column, or diagonal of the board. A set D of squares of Q(n X m) is a dominating set for Q(n X m) if every square of Q(n X m) is either in D or adjacent to a square in D. The minimum size of a dominating set of Q(n X m) is the domination number, denoted by gamma(Q(n X m)).
Less formally, gamma(Q(n X m)) is the number of queens that are necessary and sufficient to all squares of the n X m chessboard be occupied or attacked.
Chessboard 8 X 11 is of special interest, because it cannot be dominated by 5 queens, although the larger boards 9 X 11, 10 X 11 and 11 X 11 are. It is conjectured that 8 X 11 is the only counterexample of this kind of monotonicity.
LINKS
Sandor Bozoki, Table of n, a(n) for n = 1..170
S. Bozóki, P. Gál, I. Marosi, W. D. Weakley, Domination of the rectangular queen’s graph, arXiv:1606.02060 [math.CO], 2016.
S. Bozóki, P. Gál, I. Marosi, W. D. Weakley, Domination of the rectangular queen’s graph, 2016.
EXAMPLE
Table begins
m\n|1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18
--------------------------------------------------------
1 |1
2 |1 1
3 |1 1 1
4 |1 2 2 2
5 |1 2 2 2 3
6 |1 2 2 3 3 3
7 |1 2 3 3 3 4 4
8 |1 2 3 3 4 4 5 5
9 |1 2 3 4 4 4 5 5 5
10 |1 2 3 4 4 4 5 5 5 5
11 |1 2 3 4 4 5 5 6 5 5 5
12 |1 2 3 4 4 5 5 6 6 6 6 6
13 |1 2 3 4 5 5 6 6 6 7 7 7 7
14 |1 2 3 4 5 6 6 6 6 7 7 8 8 8
15 |1 2 3 4 5 6 6 6 7 7 7 8 8 8 9
16 |1 2 3 4 5 6 6 7 7 7 8 8 8 9 9 9
17 |1 2 3 4 5 6 7 7 7 8 8 8 9 9 9 9 9
18 |1 2 3 4 5 6 7 7 8 8 8 8 9 9 9 9 9 9
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Sandor Bozoki, Jun 11 2016
STATUS
approved