## Abstract

Let G be a graph with no three independent vertices. How many edges of G can be packed with edge-disjoint copies of K_{k} More specifically, let f_{k}(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 K_{k}. Turán's theorem together with Wilson's Theorem assert that f_{k}(n, m) = (1 -o(1))n^{2}/4 if m ≈ n^{2}/4. A conjecture of Erdos states that f_{3}(n, m) > (1 - o(1))n^{2}/4 for all plausible m. For any e > 0, this conjecture was open even if m ≤ n^{2}(1/4 + e). Generally, f _{k}(n, m) may be significantly smaller than n^{2}/4. Indeed, for k = 7 it is easy to show that f_{7}(n, m) ≤ 21/90 n^{2} for m ≈ 0.3n^{2}. Nevertheless, we prove the following result. For every k ≥ 3 there exists γ > 0 such that if m ≤ n^{2}(1/4 + γ) then f_{k}(n, m) ≥ (1 - o(1)))n^{2}/4. In the special case k = 3 we obtain the reasonable bound γ ≥ 10^{-4}. In particular, the above conjecture of Erdos holds whenever G has fewer than 0.250.1n^{2} edges.

Original language | English |
---|---|

Pages (from-to) | 805-817 |

Number of pages | 13 |

Journal | Combinatorics Probability and Computing |

Volume | 16 |

Issue number | 5 |

DOIs | |

State | Published - Sep 2007 |

## ASJC Scopus subject areas

- Theoretical Computer Science
- Statistics and Probability
- Computational Theory and Mathematics
- Applied Mathematics