Partitioning graphs with linear minimum degree

We prove that there exists an absolute constant $C>0$ such that, for any positive integer $k$, every graph $G$ with minimum degree at least $Ck$ admits a vertex-partition $V(G)=S\cup T$, where both $G[S]$ and $G[T]$ have minimum degree at least $k$, and every vertex in $S$ has at least $k$ neighbors in $T$. This confirms a question posted by Kühn and Osthus and is tight up to a constant factor. Our proof combines probabilistic methods with structural arguments based on Ore's Theorem on $f$-factors of bipartite graphs.