Sandwiching nearly regular degree-sequence graphs

Establish a sandwich coupling for graphs with degree sequence (d₁,…,dₙ), where every degree satisfies dᵢ=(1+o(1))d, under the condition d=ω(log n), generalizing the coupling between random d-regular graphs and binomial random graphs.

Background

The proved theorem concerns uniformly random d-regular graphs, corresponding to the constant degree sequence (d,…,d). The authors identify a natural extension to random graphs with prescribed, nearly equal degrees.

Earlier results cited in the paper had already been obtained in this broader degree-sequence setting in some regimes, but the authors explicitly state the general conjectured range and do not prove it here because they restrict their methods to the regular case.

References

They conjectured that the corresponding coupling should be possible for sequences \mathbf{d}=(d_1,\ldots,d_n) with d_i=(1+o(1))d for each i\in [n] under the same condition d=\omega(\log n).

— A proof of the Kim-Vu sandwich conjecture  (2510.20765 - Behague et al., 23 Oct 2025) in Section 1 (Introduction), paragraph following Conjecture 1