A011975 Covering numbers C(n,3,2).

1, 3, 4, 6, 7, 11, 12, 17, 19, 24, 26, 33, 35, 43, 46, 54, 57, 67, 70, 81, 85, 96, 100, 113, 117, 131, 136, 150, 155, 171, 176, 193, 199, 216, 222, 241, 247, 267, 274, 294, 301, 323, 330, 353, 361, 384, 392, 417, 425, 451, 460, 486

Offset: 3

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

Also, minimal number of triangles needed to cover every edge (and node) of a complete graph on n nodes. This problem is also known as the edge clique covering problem. - Dmitry Kamenetsky, Jan 24 2016

References

• P. J. Cameron, Combinatorics, ..., Cambridge, 1994, see p. 121.
• CRC Handbook of Combinatorial Designs, 1996, p. 262.
• W. H. Mills and R. C. Mullin, Coverings and packings, pp. 371-399 of Jeffrey H. Dinitz and D. R. Stinson, editors, Contemporary Design Theory, Wiley, 1992.

Links

• T. D. Noe, Table of n, a(n) for n = 3..1000
• Marek Cygan, Marcin Pilipczuk and Michał Pilipczuk, Known algorithms for EDGE CLIQUE COVER are probably optimal, arXiv:1203.1754 [cs.DS], 2012.
• Oliver Goldschmidt, Dorit S. Hochbaum, Cor Hurkens and Gang Yu, Approximation Algorithms for the k-Clique Covering Problem, Journal of Discrete Mathematics, volume 9, issue 3, pages 492-509, 1995, doi: 10.1137/S089548019325232X.
• D. Gordon, La Jolla Repository of Coverings
• Jenö Lehel, The minimum number of triangles covering the edges of a graph, Journal of Graph Theory, volume 13, issue 3, pages 369-384, 1989.
• Uenal Mutlu (uenalm(AT)metronet.de), Tables of coverings
• Wikipedia, Clique Cover Problem.
• Index entries for covering numbers

Crossrefs

Cf. A011976, A011977, A001839. A column of A066010. Also a column of A036838.

Programs

Maple

L := proc(v,k,t,l) local i,t1; t1 := l; for i from v-t+1 to v do t1 := ceil(t1*i/(i-(v-k))); od: t1; end; # gives Schoenheim bound L_l(v,k,t). Present sequence is L_1(n,3,2,1).

Mathematica

L[v_, k_, t_, m_] := Module[{t1 = m}, Do[t1 = Ceiling[t1*i/(i - (v - k))], {i, v - t + 1, v}]; t1]; Table[L[n, 3, 2, 1], {n, 3, 100}] (* T. D. Noe, Sep 28 2011 *)

Formula

Conjecture: G.f. ( -1-2*x-2*x^5+x^7+x^6-x^8 ) / ( (1+x+x^2)*(x^2-x+1)*(1+x)^2*(x-1)^3 ) with a(n) = a(n-1) + a(n-2) - a(n-3) + a(n-6) - a(n-7) - a(n-8) + a(n-9). - R. J. Mathar, Aug 12 2012

a(n) = ceiling((n/3)*ceiling((n-1)/2)). - Nathaniel Johnston, Jan 10 2024