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<< log n; 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 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 (0) log(2)n with 0=[1-12log23]-1 approximate to 4.82. To be precise, G contains a k-cycle factor w.h.p. if disp-formula content-type=mathematics id=cpa21613-disp-0001 kK0(n):=remvoe0log2(2n/e) disp-formula and w.h.p. does not contain one if k更多