On random irregular subgraphs

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