Let G(n, M) be the uniform random graph with n vertices and M edges. Erdos and Renyi (1960) conjectured that the limiting probability; lim(n ->infinity) Pr{G(n, n/2) is planar}; exists and is a constant strictly between 0 and 1. Luczak, Pittel and Wierman (1994) proved this conjecture, and Janson, Luczak, Knuth and Pittel (1993) gave lower and upper bounds for this probability. In this paper we determine the exact limiting probability of a random graph being planar near the critical point M = n/2. For each lambda, we find an exact analytic expression for; p(lambda) = lim(n ->infinity) Pr {G (n, n/2 (1 + lambda(n-1/3))) is planar}.; In particular, we obtain p(0) approximate to 0.99780. We extend these results to classes of graphs closed under taking minors. As an example, we show that the probability of G(n, n/2) being series-parallel converges to 0.98003. For the sake of completeness and exposition we reprove in a concise way several basic properties we need of a random graph near the critical point.

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