Bollobás–Eldridge–Catlin packing conjecture

Let $G_1$ and $G_2$ be graphs on $n$ vertices, with maximum degrees $\Delta_1$ and $\Delta_2$, respectively. Two graphs pack if there are bijections between their vertex sets such that no edge of one graph is mapped to an edge of the other.

Bollobás–Eldridge–Catlin conjecture. If

$$(\Delta_1+1)(\Delta_2+1)\le n+1,$$

then $G_1$ and $G_2$ pack.

There are some resolved cases, but the conjecture is open in general.