The evolution of the min–min random graph process

We study the following min–min random graph process G = ( G 0 , G 1 , … ) : the initial state G 0 is an empty graph on n vertices ( n even). Further, G M + 1 is obtained from G M by choosing a pair { v , w } of distinct vertices of minimum degree uniformly at random among all such pairs in G M and adding the edge { v , w } . The process may produce multiple edges. We show that G M is asymptotically almost surely disconnected if M ≤ n , and that for M = ( 1 + t ) n , 0 < t ≤ 1 2 constant, the probability that G M is connected increases from 0 to 1. Furthermore, we investigate the number X of vertices outside the giant component of G M for M = ( 1 + t ) n . For 0 < t < 1 2 constant we derive the precise limiting distribution of X . In addition, for n − 1 ln 4 n ≤ t = o ( 1 ) we show that t X converges to a gamma distribution.