A060438
Triangle T(n,k), 1 <= k <= n, giving maximal size of binary code of length n and covering radius k.
Original entry on oeis.org
1, 2, 1, 2, 2, 1, 4, 2, 2, 1, 7, 2, 2, 2, 1, 12, 4, 2, 2, 2, 1, 16, 7, 2, 2, 2, 2, 1, 32, 12, 4, 2, 2, 2, 2, 1, 62, 16, 7, 2, 2, 2, 2, 2, 1
Offset: 1
Triangle starts:
1;
2,1;
2,2,1;
4,2,2,1;
7,2,2,2,1;
...
- G. D. Cohen et al., Covering Codes, North-Holland, 1997, p. 166.
A029866
Size of minimal binary covering code of length n and covering radius 2.
Original entry on oeis.org
1, 2, 2, 2, 4, 7, 12, 16
Offset: 2
- G. D. Cohen et al., Covering Codes, North-Holland, 1997, p. 166.
- R. Bertolo, Patric R. J. Östergård and W. D. Weakley, An updated table of binary/ternary mixed covering codes, J. Combin. Designs, 12 (2004), 157-176, DOI:10.1002/jcd.20008. [a(9)=16, bounds for n>9]
- Dmitry Kamenetsky, Best known solutions for n <= 11.
- Eric Weisstein's World of Mathematics, Domination Number
- Eric Weisstein's World of Mathematics, Halved Cube Graph
- Index entries for sequences related to covering codes
Cf.
A000983 (domination number of the n-hypercube graph Q_n).
A238305
Triangle E(n,k), 1<=k<=n, giving the cardinality of optimal ternary covering codes of empty spheres of length n and radius k.
Original entry on oeis.org
2, 3, 3, 6, 4, 5, 14, 6, 5, 8, 27, 12, 6, 7, 12
Offset: 1
Triangle starts:
01: 2
02: 3 3
03: 6 4 5
04: 14 6 5 8
05: 27 12 6 7 12
...
- Jernej Azarija, M. A. Henning, S. Klavzar, (Total) Domination in Prisms, arXiv preprint arXiv:1606.08143 [math.CO], 2016.
- Jernej Azarija, S. Klavzar, Y. Rho, S. Sim, On domination-type invariants of Fibonacci cubes and hypercubes, Preprint 2016.
- Jernej Azarija, S. Klavzar, Y. Rho, S. Sim, On domination-type invariants of Fibonacci cubes and hypercubes, Ars Mathematica Contemporanea, 14 (2018) 387-395.
- Kamiel P. F. Verstraten, A generalization of the football pool problem, Master's Thesis, Tilburg University, 2014.
Related to
A060439, which has a code consisting of filled spheres instead of empty spheres.
Related to
A230014, the triangle giving the cardinality of optimal binary covering codes of empty spheres.
A230014
Triangle E(n,k), 1<=k<=n, giving the cardinality of optimal binary covering codes of empty spheres of length n and radius k.
Original entry on oeis.org
2, 2, 4, 4, 4, 8, 4, 4, 4, 16, 8, 6, 6, 8, 32, 14, 8, 6, 8, 14, 64, 24, 8, 8, 8, 8, 24, 128, 32, 16, 8, 8, 8, 16, 32, 256, 64, 24, 12, 10, 10, 12, 24, 64, 512, 124
Offset: 1
Triangle starts:
01: 2,
02: 2, 4,
03: 4, 4, 8,
04: 4, 4, 4, 16,
05: 8, 6, 6, 8, 32,
06: 14, 8, 6, 8, 14, 64,
07: 24, 8, 8, 8, 8, 24, 128,
08: 32, 16, 8, 8, 8, 16, 32, 256,
09: 64, 24, 12, 10, 10, 12, 24, 64, 512,
10: 124, ...
Related to
A060438, which has a code consisting of filled spheres instead of empty spheres.
Related to
A238305, the triangle giving the cardinality of optimal ternary covering codes of empty spheres.
The first column is equal to 2*
A000983.
A157887
The domatic number of the n-cube.
Original entry on oeis.org
1, 2, 2, 4, 4, 4, 5, 8, 8, 8
Offset: 0
Sune Kristian Jakobsen (sunejakobsen(AT)hotmail.com), Mar 08 2009
a(3)=4: The vertices of the 3-dimensional cube can be partitioned into 4 dominating sets, {000,111}, {001,110}, {010,101}, {011,100}, but not into 5. A subset of a graph is called dominating if every vertex in the graph is in the set or is a neighbor of a vertex in the set.
- Patric R. J. Östergård, A Coloring Problem in Hamming Spaces, European Journal of Combinatorics, Volume 18, Number 3, April 1997, pp. 303-309.
- Todd Trimble, Solution to POW-12: A graph coloring problem
- Eric Weisstein's World of Mathematics, Domatic Number
- Eric Weisstein's World of Mathematics, Hypercube Graph
- Wikipedia, Domatic number
A247181
Total domination number of the n-hypercube graph.
Original entry on oeis.org
2, 2, 4, 4, 8, 14, 24, 32, 64, 124
Offset: 1
a(1) = 2 since the complete graph on two vertices can only be totally dominated by taking both vertices.
- J. Azarija, M. A. Henning and S. Klavžar (Total) Domination in Prisms, arXiv:1606.08143 [math.CO], 2016.
- Jernej Azarija, S. Klavzar, Y. Rho, and S. Sim, On domination-type invariants of Fibonacci cubes and hypercubes, Preprint 2016; See Table 4.
- Jernej Azarija, S. Klavzar, Y. Rho, and S. Sim, On domination-type invariants of Fibonacci cubes and hypercubes, Ars Mathematica Contemporanea, 14 (2018) 387-395. See Table 4.
- M. Henning and A. Yeo, Total domination in graphs, Springer, 2013.
- Kamiel P. F. Verstraten, A Generalization of the Football Pool Problem, Master's Thesis, Tilburg University, 2014.
- Eric Weisstein's World of Mathematics, Hypercube Graph
- Eric Weisstein's World of Mathematics, Total Domination Number
A347555
Number of minimum dominating sets in the hypercube graph Q_n.
Original entry on oeis.org
1, 2, 6, 4, 40, 320, 2240, 240
Offset: 0
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