On two conjectures on packing of graphs
In 1978, Bollobás and Eldridge  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 . 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. © 2005 Cambridge University Press.
Combinatorics Probability and Computing
Bollobás, B., Kostochka, A., & Nakprasit, K. (2005). On two conjectures on packing of graphs. Combinatorics Probability and Computing, 14 (5-6), 723-736. https://doi.org/10.1017/S0963548305006887