### Abstract

In 1978, Bollobás and Eldridge [5] made the following two conjectures.(C1) There exists an absolute constant c > 0 such that, if k is a positive integer and $G_1$ and $G_2$ are graphs of order n such that $\Delta(G_1),\Delta(G_2)\leq n-k$ and $e(G_1),e(G_2)\leq ck n$, then the graphs $G_1$ and $G_2$ pack. (C2) For all $0 and $0, there exists an $n_0=n_0(\alpha,c)$ such that, if $G_1$ and $G_2$ are graphs of order n > n_{0} satisfying $e(G_1)\leq \alpha n$ and e(G_{2})≤ c√ n ^{3}\alpha, then the graphs $G_1$ and $G_2$ pack. Conjecture (C2) was proved by Brandt [6]. In the present paper we disprove (C1) and prove an analogue of (C2) for $1/2\leq \alpha. We also give sufficient conditions for simultaneous packings of about √n/4 sparse graphs.

Original language | English (US) |
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Pages (from-to) | 723-736 |

Number of pages | 14 |

Journal | Combinatorics Probability and Computing |

Volume | 14 |

Issue number | 5-6 |

DOIs | |

State | Published - Nov 1 2005 |

### ASJC Scopus subject areas

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

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## Cite this

*Combinatorics Probability and Computing*,

*14*(5-6), 723-736. https://doi.org/10.1017/S0963548305006887