Minimum degree k and k-connectedness usually arrive together

Abstract

Let $d,n\in N$ be such that $d=\omega \left(1\right)$, and $d\le n^{1-a}$ for some constant $a>0$. Consider a d-regular graph $G=(V,E)$ and the random graph process that starts with the empty graph $G\left(0\right)$ and at each step $G\left(i\right)$ is obtained from $G(i-1)$ by adding uniformly at random a new edge from E. We show that if G satisfies some (very) mild global edge-expansion, and an almost optimal edge-expansion of sets up to order $O(d\log n)$, then for any constant $k\in N$ in the random graph process on G, typically the hitting times of minimum degree at least k and of k-connectedness are equal. This, in particular, covers both d-regular high dimensional product graphs and pseudo-random graphs, and confirms a conjecture of Joos from 2015. We further demonstrate that this result is tight in the sense that there are d-regular n-vertex graphs with optimal edge-expansion of sets up to order $\Omega \left(d\right)$, for which the probability threshold of minimum degree at least one is different than the probability threshold of connectivity.