Behavior of the Minimum Degree Throughout the $d$-process

TL;DR

This paper analyzes how the minimum degree in a random graph evolves during the $d$-process, revealing precise probabilistic timing of degree jumps and answering a question about their distribution.

Contribution

It provides a detailed probabilistic description of the minimum degree evolution in the $d$-process, including the timing and distribution of degree jumps, extending previous understanding.

Findings

Minimum degree jumps from j to j+1 when steps remaining are about ln(n)^{d-j-1}

Distribution of steps remaining when degree j disappears converges to an exponential distribution

Distributions for different j are independent

Abstract

The $d$-process generates a graph at random by starting with an empty graph with $n$ vertices, then adding edges one at a time uniformly at random among all pairs of vertices which have degrees at most $d-1$ and are not mutually joined. We show that, in the evolution of a random graph with $n$ vertices under the $d$-process with $d$ fixed, with high probability, for each $j \in \{0,1,\dots,d-2\}$, the minimum degree jumps from $j$ to $j+1$ when the number of steps left is on the order of $\ln(n)^{d-j-1}$. This answers a question of Ruci\'nski and Wormald. More specifically, we show that, when the last vertex of degree $j$ disappears, the number of steps left divided by $\ln(n)^{d-j-1}$ converges in distribution to the exponential random variable of mean $\frac{j!}{2(d-1)!}$; furthermore, these $d-1$ distributions are independent.