On two conjectures on packing of graphs

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.