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 Bollobas and Thomason regarding connectivity. A graph G is said to be alpha-resilient with respect to a monotone increasing graph property P if for every spanning sub-graph H subset of G satisfying deg(H)(v) <= alpha deg(G)(v) for all v is an element of V(G), the graph G - H still possesses 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 >= 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 >= (1/6+ o(1))n log n, the largest connected component of G(m) is (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.