The first k-regular subgraph is large

TL;DR

This paper proves that in a random graph process, the first k-regular subgraph that appears is asymptotically large, occupying nearly the entire k-core as the number of edges increases.

Contribution

It establishes that the initial k-regular subgraph in the process is almost as large as the k-core, with precise asymptotic bounds for large k.

Findings

The first k-regular subgraph size approaches the k-core size asymptotically.

As k increases, the difference between the subgraph and the k-core diminishes.

The result holds with high probability in the random graph process.

Abstract

We prove that for sufficiently large k, there exist $0\le\sigma_k\le\eps_k\to 0$ as $k\to\infty$, such that asymptotically almost surely the first k-regular subgraph appeared in the random graph process where one edge is added at a time has size between $(1-\eps_k)|\K_k|$ and $(1-\sigma_k)|\K_k|$, where $\K_k$ denotes the $k$-core of the graph.