On resilience of connectivity in the evolution of random graphs

In this note we establish a resilience version of the classical hitting time result of Bollob'{a}s and Thomason regarding connectivity. A graph $G$ is said to be $\alpha$-resilient with respect to a monotone increasing graph property $\mathcal{P}$ if for every spanning subgraph $H\subseteq G$ satisfying ${\mathrm{d}\mathrm{e}\mathrm{g}}_H(v)\le \alpha \cdot {\mathrm{d}\mathrm{e}\mathrm{g}}_G(v)$ for all $v\in V(G)$, the graph $G-H$ still possesses $\mathcal{P}$. Let $\{G_i\}$ be the random graph process, that is a process where, starting with an empty graph on $n$ vertices $G_0$, in each step $i\ge 1$ an edge $e$ is chosen uniformly at random among the missing ones and added to the graph $G_{i-1}$. We show that the random graph process is almost surely such that starting from $m \geq (\tfrac{1}{6} + o(1)) n \log n$,thelargestconnectedcomponentof$G_m$is$(\tfrac{1}{2} - o(1))$-resilient with respect to connectivity. The result is optimal in the sense that the constants $1\mathrm{/}6$ in the number of edges and $1\mathrm{/}2$ in the resilience cannot be improved upon. We obtain similar results for $k$-connectivity.