@MISC{68onthe, author = {}, title = {ON THE NUMBER OF COMPLETE SUBGRAPHS AND CIRCUITS CONTAINED IN GRAPHS}, year = {1968} }

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Abstract

Dedicated to V. JARNiK on the occasion of his 70-th birthday. Denote by W(n; k) a graph of n vertices and k edges. Put for n- r (mod p- 1) m(n,p) = p-2 (n 2-r 2)+ ( r), 0<_n_<p-1 2(p- 1) 22 and denote by K P the complete graph of p vertices. A well known theorem of TUR.áN [6] states that every 9(n; m(n, p) + 1) contains a Kp and that this result is best possible. Thus in particular every á(2n; n 2 + 1) contains a triangle. Denote by f„(p; 1) the largest integer so that every W(n; m(n, p) + 1) contains at least fn(p; l) distinct K p 's. RADEMACHER proved that fn (3; 1) = [n/2] and I proved [1] that there exists a constant 0 < c < z so that for every (1) 1 < cn, fn (3; 1) = l [n2] and I conjectured that (1) holds for every 1 < [ n/2]. We are very far from being able to determine fn(p; 1) in general, the problem is unsolved even for p = 3 (though W. BROWN has certain plausible unpublished conjectures). NORDHAus and STEWART [4] conjectured that I proved that for l = o(n 2) lim mint"(3 ' 1)- rn 2l = 8 0 < l < (n)