Cycles in a random graph near the critical point

Let G ( n, M ) be a graph chosen at random from the family of all labelled graphs with n vertices and M ( n ) = 0.5 n + s ( n ) edges, where s 3 ( n ) n −2 →∞ but s ( n ) = o ( n ). We find the limit distribution of the length of shortest cycle contained in the largest component of G ( n, M ), as well as of the longest cycle outside it. We also describe the block structure of G ( n, M ) and derive from this result the limit probability that G ( n, M ) contains a cycle with a diagonal. Finally, we show that the probability tending to 1 as n ‐→∞ the length of the longest cycle in G ( n, M ) is of the order s 2 ( n ) /n .