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

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.