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A scaling limit for the length of the longest cycle in a sparse random digraph (2001.06481v1)
Published 17 Jan 2020 in math.CO
Abstract: We discuss the length $\vec{L}{c,n}$ of the longest directed cycle in the sparse random digraph $D{n,p},p=c/n$, $c$ constant. We show that for large $c$ there exists a function $\vec{f}(c)$ such that $\vec{L}{c,n}/n\to \vec{f}(c)$ a.s. The function $\vec{f}(c)=1-\sum{k=1}\infty p_k(c)e{-kc}$ where $p_k$ is a polynomial in $c$. We are only able to explicitly give the values $p_1,p_2$, although we could in principle compute any $p_k$.