Here is my question:

Given a graph of v vertices such that the degree of every vertex is at least n, is it necessarily true that the n-regular graph on v vertices is a subgraph of this graph?

For instance, if I have a graph with six vertices such that every vertex has degree at least 3, is it true that the 3-regular graph on six vertices is necessarily embedded in that graph?

Thanks so much! It seems as though it should be, but I would not be surprised if counterexamples existed. Any source/citation is super-appreciated but not required.