Graphs with minimum degree $\delta(G)\lt\aleph_0$

Let $ G=(V,E)$ be a graph with minimum degree $ \delta(G)=n\lt\aleph_0$ . Does $ G$ necessarily have a spanning subgraph $ G’=(V,E’)$ which also has minimum degree $ \delta(G’)=n$ and is minimal with that property?

This question is easily answered in the affirmative if $ G$ is locally finite or if $ n\le1$ . It already seems difficult for $ n=2$ , but I am not very clever and may be missing something obvious.

The question also seems to make sense for hypergraphs:

Let $ m,n\in\mathbb N$ . Let $ E$ be a family of sets, each of cardinality at most $ m$ . If $ E$ is an $ n$ -cover of a set $ V$ (each element of $ V$ is in at least $ n$ elements of $ E$ ), does $ E$ contain a minimal $ n$ -cover of $ V$ ?

I would expect such simple questions to have been asked and answered 100 years ago.

Where are these questions considered in the literature?