Suppose that a process begins with n isolated vertices, to which edges are added randomly one by one so that the maximum degree of the induced graph is always at most 2. In a previous article, the authors showed that as $n \to \infty$, with probability tending to 1, the result of this process is a graph with n edges. The number of l-cycles in this graph is shown to be asymptotically Poisson $(1 \geq 3)$, and other aspects of this random graph model are studied.
"Random graph processes with maximum degree $2$." Ann. Appl. Probab. 7 (1) 183 - 199, February 1997. https://doi.org/10.1214/aoap/1034625259