For a graph G and a fixed integer k ≥ 3, let νk(G) denote the maximum number of pairwise edge-disjoint copies of Kk in G. For a constant c, let η(k, c) be the infimum over all constants γ such that any graph G of order n and minimum degree at least cn has νk((G) = γn2(1 - on(1)). By Turán's theorem, η(k, c) = 0 if c ≤ 1 - 1/(k - 1) and by Wilson's theorem, η(k, c) ≥ 1/(k2 - k) as c → 1. We prove that for any 1 > c > 1-1/(k-1), η(k, c) ≥ c/2- ((k 2)-1)ck-1 2Π k-2i=1 ((i+1)c-i)+2((k 2)-1)ck-2, while it is conjectured that η(k, c) = c/(k2 - k) if c ≥ k/(k +1) and η(k, c) = c/2 - (k - 2)/(2k - 2) if k/(k + 1) > c > 1 - 1/(k - 1). The case k = 3 is of particular interest. In this case the bound states that for any 1 > c > 1/2, η(3, c) = c/2 - c2 4c-1. By further analyzing the case k = 3 we obtain the improved lower bound η(3, c) ≥ (12 c2-5 c+2- √240 c4-216 c3+73 c 2-20 c+4)(2 c-1)/ 32(1-c)c . This bound is always at most within a fraction of (20 - √238)/6 > 0.762 of the conjectured value, which is η(3, c) = c/6 for c = 3/4 and η(3, c) = c/2 - 1/4 if 3/4 > c > 1/2. Our main tool is an analysis of the value of a natural fractional relaxation of the problem.
ASJC Scopus subject areas
- Mathematics (all)