A295428 -id:A295428 - cp's OEIS frontend

A303334 Number of dominating sets in the n X n torus grid graph.

421, 45707, 18935741, 30147126791, 183429997068809, 4264383011657313355, 378801055723829891830261, 128572687866388429165521180651, 166751578049943666873090557914876017, 826369316231187306403443156508234313658719, 15648091232010974513543383340423707992718009437149

Offset: 3

Andrew Howroyd, Apr 21 2018

nonn

Stephan Mertens, Table of n, a(n) for n = 3..17
Stephan Mertens, Domination Polynomials of the Grid, the Cylinder, the Torus, and the King Graph, arXiv:2408.08053 [math.CO], Aug 2024.
Eric Weisstein's World of Mathematics, Dominating Set
Eric Weisstein's World of Mathematics, Torus Grid Graph

Cf. A295428, A298106, A303210.

a(9) and beyond from Stephan Mertens, Aug 18 2024

A094087 Domination number of the Cartesian product of two n-cycles.

Original entry on oeis.org

1, 2, 3, 4, 5, 8, 12, 16, 18, 20, 27, 32, 38, 42, 45, 56, 64, 71, 76, 80, 95, 104, 114, 120, 125, 144, 155

Offset: 1

Richard Bean, May 01 2004

1/5 <= a(n)/n^2 <= 1/4 for n >= 4; it is conjectured that a(5n-1) = 5*n^2 - n and a(5n+1) = 5n^2 + 4n - 1 for n >= 1. - Richard Bean, Sep 08 2006 [Assadian proves that the both conjectured formulas give the upper bounds. - Andrey Zabolotskiy, Dec 23 2019]

The Cartesian product of two cycles is also called the torus grid graph. - Andrew Howroyd, Feb 29 2020

Navid Assadian, Dominating Sets of the Cartesian Products of Cycles, M. Sc. project, University of Victoria, 2019.
S. Klavžar and N. Seifter, Dominating Cartesian products of cycles, Discrete Applied Mathematics, Vol. 59 (1995), no. 2, pp. 129-136.
Stephan Mertens, Domination Polynomials of the Grid, the Cylinder, the Torus, and the King Graph, arXiv:2408.08053 [math.CO], 2024. See p. 15.
Zehui Shao, Jin Xu, S. M. Sheikholeslami, and Shaohui Wang, The Domination Complexity and Related Extremal Values of Large 3D Torus, Complexity, 2018, 3041426.
Eric Weisstein's World of Mathematics, Domination Number
Eric Weisstein's World of Mathematics, Torus Grid Graph

Cf. A295428, A302406, A303334.

a(5n) = 5n^2. - Richard Bean, Jun 08 2006

More terms from Richard Bean, Sep 08 2006
a(22) from Richard Bean, Jul 24 2018
a(23)-a(24) from Shao et al. added by Andrey Zabolotskiy, Dec 23 2019
a(25)-a(27) from Richard Bean, Apr 03 2022

cp's OEIS Frontend

A303334 Number of dominating sets in the n X n torus grid graph.

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