A scaling limit for the length of the longest cycle in a sparse random digraph

Abstract

We discuss the length ⃗ 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 functionwhere 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 .

Cycle lengths in sparse random graphs

“…Anastos and Frieze [1] then proved that this holds when c > C 0 for some absolute constant C 0 , and further identified the limit f (c). The analogous result for (n, p) was thereafter obtained by the same authors in [2]. For c = 1 + o(1) outside the critical window, L max (G) is known up to constant factors [20]; see Remark 4.6.…”

“…The analogous statement for G ∼ (n, p) follows from Corollary 4.4 in exactly the same manner as argued above, except that now, rather than relying on [1], we appeal to the sequel by the same authors [2] for the fact that there exists some absolute C 0 > 0 such that, if G ∼ (n, p) with p = c∕n for fixed c > C 0 then L max (G)∕n → f (c) in probability for some (non-decreasing) function f (c). Finally, the joint law of short cycles is again that of asymptotically independent Poisson random variables (e.g., via the same method-of-moments argument referenced above, and stated for arbitrary strictly balanced graphs in [5,Theorem 4.8]), yet now the automorphism group of a k-cycle in the directed graph G has order k rather than 2k.…”

“…$$ Combining this with () shows that $$$ \mathbb{P}\left(\left\llbracket \ell, \left(1-\varepsilon \right){L}_{\mathrm{max}}(G)\right\rrbracket \subset \mathcal{L}(G)\right) $$$ is within $$$ {\varepsilon}^{\prime }+o(1) $$$ of $$$ \theta \left(c,\ell \right) $$$, as required. The analogous statement for follows from Corollary 4.4 in exactly the same manner as argued above, except that now, rather than relying on [1], we appeal to the sequel by the same authors [2] for the fact that there exists some absolute $$$ {C}_0>0 $$$ such that, if with $$$ p=c/n $$$ for fixed $$$ c>{C}_0 $$$ then $$$ {L}_{\mathrm{max}}(G)/n\to f(c) $$$ in probability for some (non‐decreasing) function $$$ f(c) $$$. Finally, the joint law of short cycles is again that of asymptotically independent Poisson random variables (e.g., via the same method‐of‐moments argument referenced above, and stated for arbitrary strictly balanced graphs in [5, Theorem 4.8]), yet now the automorphism group of a k ‐cycle in the dir...…”

“…by [23], giving the constant 𝛾 0 . The upper bound 𝛾 1 can be obtained by observing that a Hamilton cycle in  traverses a 2 3 fraction of the kernel's edges. Thus, taking the longest 2 3 of the paths replacing the edges of , combined with the classical representation of order statistics for i.i.d.…”

Depth first exploration of a configuration model

“…However, a main advantage of our bound is that it is given by an explicit construction in linear time, which is not the case for the regular graphs setting. For additional details and references on the Erdős-Rényi setting, we refer to the survey [14] and to the article [3].…”