On the Resilience of Long Cycles in Random Graphs

In this paper we determine the local and global resilience of random graphs Gn,p (p n−1) with respect to the property of containing a cycle of length at least (1 − α)n. Roughly speaking, given α > 0, we determine the smallest rg(G,α) with the property that almost surely every subgraph of G = Gn,p having more than rg(G,α)|E(G)| edges contains a cycle of length at least (1−α)n (global resilience). We also obtain, for α < 1/2, the smallest rl(G,α) such that any H ⊆ G having degH(v) larger than rl(G,α) degG(v) for all v ∈ V (G) contains a cycle of length at least (1 − α)n (local resilience). The results above are in fact proved in the more general setting of pseudorandom graphs. Supported by a CAPES–Fulbright scholarship. Partially supported by FAPESP and CNPq through a Temático–ProNEx project (Proc. FAPESP 2003/09925–5) and by CNPq (Proc. 306334/2004–6 and 479882/2004–5). the electronic journal of combinatorics 15 (2008), #R32 1