On random irregular subgraphs

Let $G$ be a $d$-regular graph on $n$ vertices. Frieze, Gould, Karoński, and Pfender began the study of the following random spanning subgraph model $H=H(G)$. Assign independently to each vertex $v$ of $G$ a uniform random number $x(v)\in [0,1]$, and an edge $(u,v)$ of $G$ is an edge of $H$ if and only if $x(u)+x(v)\ge 1$. Addressing a problem of Alon and Wei, we prove that if $d=o(n/{(\log n)}^{12})$, then with high probability, for each nonnegative integer $k\le d$, there are $(1+o(1))n/(d+1)$ vertices of degree $k$ in $H$.