Cycles of given lengths in unicyclic components in sparse random graphs

2020/03/07 by Marc Noy, Vonjy Rasendrahasina, Noy, Marc +5
Mathematics · #Combinatorics (math.CO) #FOS: Mathematics #math.CO

paper · pdf · doi:10.48550/arxiv.2003.03552

arxiv created 2020/12/28 · arxiv updated 2020/12/29

Abstract

Let L be subset of \3,4,…\ 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=(n)/(2)(1+μn^-1/3) with μ=O(1). Depending on the regime and a condition involving the series ∑_{l ∈ L} (z^l)/(2l), we obtain in the limit either a Poisson or a normal distribution as n→∞.

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