Nearly-uniform degree distributions in spanning subgraphs

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.