On the growth of components with non-fixed excesses

Denote by an l-component a connected graph with l edges more than vertices. We prove that the expected number of creations of (l + 1)-component, by means of adding a new edge to an l-component in a randomly growing graph with n vertices, tends to 1 as l, n tends to ∞ but with l = o(n1/4). We also show, under the same conditions on l and n, that the expected number of vertices that ever belong to an l-component is ∼ (12l)1/3n2/3.