Asymptotic rate for fixed odd regularity

Determine whether, for every fixed odd integer k≥3, the maximum proportion δ_k(r) of vertices that may be omitted by a k-regular subgraph of an r-regular graph satisfies δ_k(r)=Θ_k(r^{-1}) as r tends to infinity.

Background

The paper defines δ_k(r) as the largest proportion of vertices that may remain uncovered by a k-regular subgraph in an r-regular graph. For odd k and odd r, the construction discussed in the paper yields the lower bound δ_k(r)≥(r-k)/(r+1)^2, which is of order r^{-1} for every fixed odd k.

The paper's general upper bound is O(k/\sqrt r), so it does not establish the conjectured inverse-linear rate for fixed odd k≥3. The conjecture seeks matching Θ_k(r^{-1}) behavior as r grows, extending the sharp asymptotic behavior known for k=1.