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Cycles of given lengths in unicyclic components in sparse random graphs (2003.03552v2)
Published 7 Mar 2020 in math.CO
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 $\sum_{l \in L} \frac{zl}{2l}$, we obtain in the limit either a Poisson or a normal distribution as $n\to\infty$.