## Abstract

Let*V*
_{
n
}={1, 2, ...,*n*} and*e*
_{1},*e*
_{2}, ...,*e*
_{
N
},*N*=\(\left( {\begin{array}{*{20}c} n \\ 2 \\ \end{array} } \right)\) be a random permutation of*V*
_{n}
^{(2)}. Let*E*
_{t}={*e*
_{1},*e*
_{2}, ...,*e*
_{t}} and*G*
_{t}=(*V*
_{
n
},*E*
_{
t
}). If*Π* is a monotone graph property then the hitting time*τ(Π)* for*Π* is defined by*τ=τ(Π)*=min {*t:G*
_{
t
}
*∈Π*}. Suppose now that*G*
_{τ} starts to deteriorate i.e. loses edges in order of*age, e*
_{1},*e*
_{2}, .... We introduce the idea of the*survival* time τ =τ′(*Π*) defined by τt = max {u:(*V*
_{n}, {*e*
_{u},*e*
_{u+1}, ...,*e*
_{
T
}
*}*) ∈*Π*}. We study in particular the case where*Π* is*k*-connectivity. We show that

i.e.*τ′/n* is asymptotically negative exponentially distributed with mean 1/2.

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### Cite this article

Frieze, A.M., Frieze, A.M. Survival time of a random graph.
*Combinatorica* **9, **133–143 (1989). https://doi.org/10.1007/BF02124675

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### AMS subject classification (1980)

- 05 C 80