SOLVED
Does every regular graph of degree $4$ contain a regular subgraph of degree $3$? Is there any $r$ such that every regular graph of degree $r$ must contain a regular subgraph of degree $3$?
A problem of Berge (or Berge and Sauer). Alon, Friedland, and Kalai
[AFK84] proved that every $4$-regular graph plus an edge contains a $3$-regular subgraph, and hence in particular every $r$-regular graph with $r\geq 5$ contains a $3$-regular subgraph.
The answer is yes, proved by Tashkinov [Ta82].
