Suppose, I have an $r$-regular graph with $r>2$. Does there exist a subgraph with $r$ nodes such that each node in the subgraph has at least $r-2$ connections with other nodes in the subgraph?
For some graphs, I have checked that this is true. For a $3$-regular graph a line with three nodes, for $4$-regular graph a cycle of $4$ nodes; for a $5$-regular graph and for $6$-regular graph this is true. Is this in general true? Or does it require more conditions to be true?