On the Resilience of Long Cycles in Random Graphs

Abstract

In this paper we determine the local and global resilience of random graphs $G_{n, p}$ ($p \gg n^{-1}$) with respect to the property of containing a cycle of length at least $(1-\alpha)n$. Roughly speaking, given $\alpha > 0$, we determine the smallest $r_g(G, \alpha)$ with the property that almost surely every subgraph of $G = G_{n, p}$ having more than $r_g(G, \alpha) |E(G)|$ edges contains a cycle of length at least $(1 - \alpha) n$ (global resilience). We also obtain, for $\alpha < 1/2$, the smallest $r_l(G, \alpha)$ such that any $H \subseteq G$ having $\deg_H(v)$ larger than $r_l(G, \alpha) \deg_G(v)$ for all $v \in V(G)$ contains a cycle of length at least $(1 - \alpha) n$ (local resilience). The results above are in fact proved in the more general setting of pseudorandom graphs.