A note on long cycles in sparse random graphs (Q6042104)

Summary: Let \(L_{c,n}\) denote the size of the longest cycle in \(G(n,{c}/{n}), c>1\) constant. We show that there exists a continuous function \(f(c)\) such that \(L_{c,n}/n \to f(c)\) a.s. for \(c\geqslant 20\), thus extending a result of \textit{M. Anastos} and \textit{A. Frieze} [J. Comb. Theory, Ser. B 148, 184--208 (2021; Zbl 1459.05059)] to smaller values of \(c\). Thereafter, for \(c\geqslant 20\), we determine the limit of the probability that \(G(n,c/n)\) contains cycles of every length between the length of its shortest and its longest cycles as \(n\to \infty \).