A274138 Triangle read by rows: Domination number for rectangular queens' graph Q(n,m), 1 <= n <= m.

1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 1, 2, 2, 2, 3, 1, 2, 2, 3, 3, 3, 1, 2, 3, 3, 3, 4, 4, 1, 2, 3, 3, 4, 4, 5, 5, 1, 2, 3, 4, 4, 4, 5, 5, 5, 1, 2, 3, 4, 4, 4, 5, 5, 5, 5, 1, 2, 3, 4, 4, 5, 5, 6, 5, 5, 5, 1, 2, 3, 4, 4, 5, 5, 6, 6, 6, 6, 6, 1, 2, 3, 4, 5, 5, 6, 6, 6, 7, 7, 7, 7, 1, 2, 3, 4, 5, 6, 6, 6, 6, 7, 7, 8, 8, 8

Offset: 1

Author

Sandor Bozoki, Jun 11 2016

Keywords

• nonn
• tabl

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.

Examples

			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
		

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.

Crossrefs

Diagonal elements are in A075458: Domination number for queens' graph Q(n).