Nearly Spanning Regular Subgraphs

Alon and Mubayi asked whether, for every integer $k\ge 1$ and every $\epsilon >0$, there exists $r_0=r_0(k,\epsilon )$ such that every $r$-regular graph on $n$ vertices with $r\ge r_0$ contains a $k$-regular subgraph covering at least $(1-\epsilon )n$ vertices. Previously, the conjecture was known for $k\in \{1,2\}$ and for $k$ and $r$ both even. We answer this question affirmatively for general $k$ and $r$ with $\epsilon =O_k(r^{-1\mathrm{/}2})$. For $k=2$ and every odd $r\ge 3$, we show that $\epsilon =1\mathrm{/}(r^2-3)$ suffices which is best possible.