On random irregular subgraphs

TL;DR

This paper analyzes a random subgraph model of a $d$-regular graph, showing that under certain degree conditions, the degree distribution in the subgraph is approximately uniform across degrees.

Contribution

It proves that for sparse enough regular graphs, the degree distribution in the random subgraph is nearly uniform across all degrees up to $d$, addressing a problem posed by Alon and Wei.

Findings

Degree distribution in the subgraph is approximately uniform for degrees up to $d$.

The result holds with high probability for $d = o(n/( ext{log } n)^{12})$.

Number of vertices with each degree $k$ is about $n/(d+1)$.

Abstract

Let $G$ be a $d$-regular graph on $n$ vertices. Frieze, Gould, Karo\'nski 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 $x(u)+x(v) \geq 1$. 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 $k \leq d$, there are $(1+o(1))n/(d+1)$ vertices of degree $k$ in $H$.