On resilience of connectivity in the evolution of random graphs

TL;DR

This paper proves that in the evolution of random graphs, the largest component remains highly resilient to edge removal, ensuring connectivity even after significant adversarial deletions, with optimal constants.

Contribution

The paper establishes a resilience version of the hitting time for connectivity in random graphs, providing optimal bounds for resilience and extending results to k-connectivity.

Findings

Largest component is $(rac{1}{2}-o(1))$-resilient after $(rac{1}{6}+o(1)) n \log n$ edges

Resilience bounds are proven to be optimal

Results extend to k-connectivity

Abstract

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 $\mathrm{deg}_H(v) \leq \alpha \cdot \mathrm{deg}_G(v)$ 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 $i \geq 1$ 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 $m \geq (\tfrac{1}{6} + o(1)) n \log n$, the largest connected component of $G_m$ is $(\tfrac{1}{2} - o(1))$-resilient with respect to connectivity. The result is optimal in the sense that the constants $1/6$ in the number of edges and $1/2$ in the resilience cannot be improved upon. We obtain similar results for $k$-connectivity.