A075561 -id:A075561 - cp's OEIS frontend

A229803 Domination number for rook graph HR(n) on a triangular board of hexagonal cells. The rook can move along any row of adjacent cells, in any of the three directions.

1, 1, 2, 2, 3, 3, 3, 4, 4, 5, 5, 6, 6, 7, 7, 8, 8, 9, 9, 9, 10, 10, 11, 11

Offset: 1

Stan Wagon, Sep 29 2013

The value for HR(20) was obtained by Rob Pratt, Sep 29 2013, using integer-linear programming.

			For HR(7), the graph can be dominated by the three vertices 6, 11, 26, where we count down from the top.
This graph was called the Queen graph in the DeMaio and Tran paper, but the moves are those of a rook in the classic hexagonal chess game.

J. Konhauser, D. Velleman, S. Wagon, Which Way Did the Bicycle Go? Washington, DC, Math. Assoc. of America, 1996, pp. 169-172

William Herbert Bird, Computational methods for domination problems, University of Victoria, 2017. See Table 2.4 on p. 32.
J. DeMaio and H. L. Tran, Domination and independence on a triangular honeycomb chessboard, Coll. Math. J. 44 (2013) 307-314.
Stan Wagon, Graph Theory Problems from Hexagonal and Traditional Chess, The College Mathematics Journal, Vol. 45, No. 4, September 2014, pp. 278-287

Cf. A075458, A075324, A075561, A006075.

a(21)-a(24) from Bird added by Andrey Zabolotskiy, Sep 03 2021

A362016 Maximal number of unmarked cells with at least 3 marked neighboring cells in the n X n kings' graph.

Original entry on oeis.org

0, 1, 4, 8, 13, 20, 28, 38, 50, 61, 75, 90, 108, 124, 139

Offset: 1

Tomas Rigaux, Apr 04 2023

The value of r = lim sup a(n) / n^2 is in the half-open interval [2/3, 8/11).

It appears from the computed terms that r = 2/3.

			a(2) = 1, as the only pattern is
  .X
  XX
a(9) = 50, with a similar pattern to prove that r >= 2/3:
  X.......X
  .XXXXXXX.
  X.......X
  .........
  XXXXXXXXX
  .........
  X.......X
  .XXXXXXX.
  X.......X
a(10) = 61, and a pattern that reaches that is
  X..X...X..
  XX.X.X.X.X
  ..........
  .X.XX.X.XX
  XX....X...
  ....X....X
  X.XX..XX.X
  X....X....
  ...X....XX
  XX.X.XX.X.

Tomas Rigaux, Python program

If we only want 1 marked neighbor, we get n^2 - A075561(n).

cp's OEIS Frontend

A229803 Domination number for rook graph HR(n) on a triangular board of hexagonal cells. The rook can move along any row of adjacent cells, in any of the three directions.

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