Cycle lengths in sparse random graphs

Abstract

We study the set L (G) of lengths of all cycles that appear in a random d-regular G on n vertices for a fixed d ≥ 3, as well as in Erdős-Rényi random graphs on n vertices with a fixed average degree c > 1. Fundamental results on the distribution of cycle counts in these models were established in the 1980's and early 1990's, with a focus on the extreme lengths: cycles of fixed length, and cycles of length linear in n.Here we derive, for a random d-regular graph, the limiting probability that L (G) simultaneously contains the entire range {ℓ, . . . , n} for ℓ ≥ 3, as an explicit expression θ ℓ = θ ℓ (d) ∈ (0, 1) which goes to 1 as ℓ → ∞.For the random graph G(n, p) with p = c/n, where c ≥ C 0 for some absolute constant C 0 , we show the analogous result for the range {ℓ, . . . , (1 − o(1))Lmax(G)}, where Lmax is the length of a longest cycle in G. The limiting probability for G(n, p) coincides with θ ℓ from the d-regular case when c is the integer d − 1. In addition, for the directed random graph D(n, p) we show results analogous to those on G(n, p), and for both models we find an interval of cε 2 n consecutive cycle lengths in the slightly supercritical regime p = 1+ε n .