Cycle Factors and Renewal Theory

For which values of k does a uniformly chosen 3-regular graph G on n vertices typically contain n/k vertex-disjoint k-cycles (a k-cycle factor)? To date, this has been answered for k = n and for k logn; the former, the Hamiltonicity problem, was finally answered in the affirmative by Robinson and Wormald in 1992, while the answer in the latter case is negative since with high probability (w.h.p.) most vertices do not lie on k-cycles. A major role in our study of this problem is played by renewal processes without replacement, where one wishes to estimate the probability that in a uniform permutation of a given a set of positive integers, the partial sums hit a designated target integer. Using sharp tail estimates for these renewal processes, which may be of independent interest, we settle the cycle factor problem completely: the “threshold” for a k-cycle factor in G as above is κ0 log2 n with κ0 = [1− 1 2 log2 3] −1 ≈ 4.82. Precisely, G contains a k-cycle factor w.h.p. if k ≥ K0(n) := dκ0 log2(2n/e)e and w.h.p. does not contain one if k < K0(n) − log n/n. Thus, for most values of n the threshold concentrates on the single integer K0(n). As a byproduct, we confirm the “Comb Conjecture,” an old problem concerning the embedding of certain spanning trees in the random graph G(n, p).