Let $G$ be a simple graph such that
- $|V(G)|$ is even
- $\delta(G) \gt \tfrac12|V(G)|+3$ where $\delta(G)$ is the minimum degree of a vertex in $G$.
The question is to prove there exists a $5$-regular spanning subgraph of $G$.
I'm trying to figure this out for at least 10 hours, but no avail. :(
The only thing I've managed to do is prove that $|V(G)| \ge 8$ and that $G$ is Hamiltonian (via Dirac's theorem).
Any ideas?