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{\bf Domingos Dellamonica Jr, Yoshiharu Kohayakawa, Martin Marciniszyn and Angelika Steger }
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{\bf On the Resilience of Long Cycles in Random Graphs}
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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.
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