Packing Cliques in Graphs with Independence Number 2

Let G be a graph with no three independent vertices. How many edges of G can be packed with edge-disjoint copies of Kk? More specifically, let fk(n, m) be the largest integer t such that for any graph with n vertices, m edges, and independence number 2, at least t edges can be packed with edge-disjoint copies of Kk. Tur´an’s Theorem together with Wilson’s Theorem assert that fk(n, m) = (1−o(1))n2 4 if m ≈ n2 4 . A conjecture of Erd˝os states that f3(n, m) ≥ (1−o(1))n2 4 for all plausible m. For any > 0, this conjecture was still open even if m ≤ n2 (1 4 + ). Generally, fk(n, m) may be significantly smaller than n2 4 . Already for k = 7 it is easy to show that f7(n, m) ≤ 21 90 n2 for m ≈ 0.3n2 . Nevertheless, we prove the following result. For every k ≥ 3 there exists γ > 0 so that if m ≤ n2 (1 4 + γ) then fk(n, m) ≥ (1 − o(1))n2 4 . In the special case k = 3 we obtain the reasonable bound γ ≥ 10−4 . In particular, the above conjecture of Erd�...

• Type: Journal
• Year: 2007
• Where: CPC
• Authors: Raphael Yuster
• Keywords: CPC 2007, Edge-disjoint Copies, Independent Vertices, Largest Integer