Cycles of given lengths in unicyclic components in sparse random graphs

Abstract

Let $L$ be subset of $\{3,4,\dots\}$ and let $X_{n,M}^{(L)}$ be the number of cycles belonging to unicyclic components whose length is in $L$ in the random graph $G(n,M)$. We find the limiting distribution of $X_{n,M}^{(L)}$ in the subcritical regime $M=cn$ with $c<1/2$ and the critical regime $M=\frac{n}{2}\left(1+\mu n^{-1/3}\right)$ with $\mu=O(1)$. Depending on the regime and a condition involving the series $\sum_{l \in L} \frac{z^l}{2l}$, we obtain in the limit either a Poisson or a normal distribution as $n\to\infty$.