OFFSET
3,2
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.
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
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 *)
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
STATUS
approved