# On two conjectures on packing of graphs

Béla Bollobás, Alexandr Kostochka, Kittikorn Nakprasit

Research output: Contribution to journalArticle

### 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 > n0 satisfying $e(G_1)\leq \alpha n$ and e(G2)≤ 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) 723-736 14 Combinatorics Probability and Computing 14 5-6 https://doi.org/10.1017/S0963548305006887 Published - Nov 1 2005

### ASJC Scopus subject areas

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