Asymptotic rate for fixed even regularity

Determine whether, for every fixed even integer k≥4, 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^{-2}) as r tends to infinity through odd integers.

Background

For an r-regular graph G, the quantity δ_k(r) is defined as the supremum over all such graphs of the proportion of vertices not covered by a largest k-regular subgraph. The paper proves the exact value δ_2(r)=1/(r^2-3) for every odd r≥3 and gives a construction showing δ_k(r)≥1/(r^2-3) for every even k. These results motivate the conjectured inverse-square asymptotic rate for every fixed even k≥4 when r is odd and grows without bound.

The general theorem established in the paper gives only the upper bound δ_k(r)=O(k/\sqrt r), while additional arguments mentioned by the authors improve this for fixed even k to O_k(log r/r), still short of the conjectured Θ_k(r^{-2}) rate. Thus the conjecture asks for matching inverse-square upper and lower bounds in this parity regime.