Perhaps the most basic property of a graph is that of being connected. Thus it is not surprising that the study of connectedness of a r.g. has a vast literature. In fact, for fear of upsetting the balance of the book we cannot attempt to give an account of all the results in the area.

Appropriately, the very first random graph paper of Erdős and Rényi (1959) is devoted to the problem of connectedness, and so are two other of the earliest papers on r.gs: Gilbert (1959) and Austin et al. (1959). Erdős and Rényi proved that (n/2) log n is a sharp threshold function for connectedness. Gilbert gave recurrence formulae for the probability of connectedness of Gp (see Exx. 1 and 2). S. A. Stepanov (1969a, 1970a, b) and Kovalenko (1971) extended results of Erdős and Rényi to the model G{n, (pij)}, and Kelmans (1967a) extended the recurrence formulae of Gilbert. Other extensions are due to Ivchenko (1973b, 1975), Ivchenko and Medvedev (1973), Kordecki (1973) and Kovalenko (1975). In §1 we shall present some of these results in the context of the evolution of random graphs.

We know from Chapter 6 that a.e. graph process is such that a giant component appears shortly after time n/2, and the number of vertices not on the giant component decreases exponentially.