Let $G$ be a graph, and minimum degree $\delta(G)\geq k$, does $G$ contain a $k$-regular subgraph?
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Pick a square with the diagonals, as a graph on FIVE vertices, where the vertices are the four corners and the intersection of the diagonals.
Then $\delta(G)\geq 3$ but it is trivial to see that $G$ contains no $3$-regular subgraph.