On the existence of a factor of degree one of a connected random graph

In a series of papers (see [1], [2], [3]) we have considered the structure of a random graph T,,, N obtained as follows : we select at random N edges among the n (n possible edges connecting n given points so that each of the l N l possible (n) -> choices has the same probability 12 In [1] we have proved that if n and N tend both to + so that (0-1) N= 'z n log n+cn+o(n) then denoting by C o the class of connected graphs and by Pn , N (A) the probability that the graph T,, .,, belongs to the class A, we have (0.2) lien P,,, N(Co ) = ee 1-N-" ti log n+en+o(n)