Nearly-uniform degree distributions in spanning subgraphs

Richard Montgomery; Alexey Pokrovskiy; Benny Sudakov

Nearly-uniform degree distributions in spanning subgraphs

Combinatorics 2026-06-29 v1

Authors: Richard Montgomery , Alexey Pokrovskiy , Benny Sudakov

Abstract

We show that, when $d=o(n)$ , every $d$ -regular $n$ -vertex graph contains a spanning subgraph whose degree distribution is nearly uniform, i.e., for each $0\leq i\leq d$ , there are $(1+o(1))n/(d+1)$ vertices with degree $i$ . This proves a conjecture of Alon and Wei on irregular subgraphs and strengthens a previous result of Fox, Luo and Pham.

Cite

@article{arxiv.2606.30612,
  title  = {Nearly-uniform degree distributions in spanning subgraphs},
  author = {Richard Montgomery and Alexey Pokrovskiy and Benny Sudakov},
  journal= {arXiv preprint arXiv:2606.30612},
  year   = {2026}
}