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

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

Author

Richard Bean, May 01 2004

Keywords

• nonn
• more

Comments

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

Links

• 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

Crossrefs

Cf. A295428, A302406, A303334.

Formula

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

Extensions

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