2 problems

Let $r\geq 2$, and let $G$ be an $((r-1)n+1)$-regular graph on $rn$ vertices. A subset $S\subseteq V(G)$ is said to induce a $K_r$-factor when $G[S]$ contains a collection of verte…

Let $G$ be an $(n+1)$-regular graph on $2n$ vertices. A subset $S\subseteq V(G)$ is called Hamiltonian when the induced subgraph $G[S]$ contains a Hamilton cycle. Erdős–Faudree con…