Dynamic Erd\"os- R\'enyi random graph with forbidden degree

TL;DR

This paper investigates a modified Erd"os-Rényi graph process with a forbidden degree, analyzing how the presence of a giant component depends on the forbidden degree and time, revealing a phase transition at degree 4.

Contribution

It introduces and analyzes a variant of the Erd"os-Rényi process with a forbidden degree, establishing the conditions for the emergence of a giant component.

Findings

Giant component appears for forbidden degree k>4

No giant component for k<4

Local limit analysis provides insights at the critical case k=4

Abstract

As suggested by Itai Benjamini, we introduced a variant of the Erd\"os- R\'enyi random graph process with a forbidden degree $k$, in which every edge adjacent to a vertex $v$ is removed when the degree of $v$ reaches $k$ (but the removed edges may very well be added again later). We study the existence of a giant component, depending on the forbidden degree $k$ and the time parameter $t$. We prove that for $k\textgreater{}4$ a giant component appears at some point, while for $k\textless{}4$, a giant component never occurs. The main tool of our study is the local limit of the random graph process: it provides useful information about the cases $k\textgreater{}4$, but also the threshold case k=4.