The evolution of the min-min random graph process

We study the following min-min random graph process G = (G0, G1, . . .): the initial state G0 is an empty graph on n vertices (n even). Further, GM+1 is obtained from GM by choosing a pair {v, w} of distinct vertices of minimum degree uniformly at random among all such pairs in GM and adding the edge {v, w}. The process may produce multiple edges. We show that GM is asymptotically almost surely disconnected if M ≤ n, and that for M = (1+ t)n, 0 < t ≤ 1 2 constant, the probability that GM is connected increases from 0 to 1. Furthermore, we investigate the number X of vertices outside the giant component of GM for M = (1 + t)n. For 0 < t < 1 2 constant we derive the precise limiting distribution of X. In addition, for n ln n ≤ t = o(1) we show that tX converges to a gamma distribution.