Aharoni's transversal Dirac conjecture

Let $\mathcal{G} = \{G_1, \ldots, G_n\}$ be a collection of graphs with common vertex set $V$ of size $n$. Define $\delta(\mathcal{G})=\min\{\delta(G_i):1\leq i\leq n\}$, where $\delta(G_i)$ is the minimum degree of $G_i$. A transversal Hamilton cycle is a Hamilton cycle whose edges can be assigned distinct representatives from the graphs $G_1,\ldots,G_n$, with the edge assigned to $G_i$ belonging to $G_i$. Aharoni's transversal Dirac conjecture. If

$$\delta(\mathcal{G})\geq \frac{n}{2},$$

then $\mathcal{G}$ contains a transversal Hamilton cycle. This conjecture extends Dirac's Hamilton-cycle theorem to graph collections; it was solved asymptotically by Cheng, Wang and Zhao and completely confirmed by Joos and Kim.