An improved lower bound on the length of the longest cycle in random graphs

Abstract

We provide a new lower bound on the length of the longest cycle of the binomial random graph G ∼ G(n, (1 + )/n) that holds w.h.p. for all = (n) such that 3 n → ∞. In the case ≤ 0 for some sufficiently small constant 0 , this bound is equal to 1.581 2 n which improves upon the current best lower bound of 4 2 n/3 due to Luczak.

Heavy and Light Paths and Hamilton Cycles

“…At each step, we observe the remaining edges of the last vertex in the path we constructed thus far (there are at most two such edges, since we assumed our graph has maximum degree 3). If we have two choices allowing us to complete a path of length k, we can choose the maximum between them, or the minimum between them, having an expected weight 3 2 and 1 2 , respectively. Otherwise, we must choose the only edge possible allowing us to complete a path of length k, having an expected weight 1.…”

“…Ajtai, Komlós, and Szemerédi [1] proved that a longest cycle in the giant is whp of order Θ(ǫ 2 )n (see also [12] for a simple proof of this fact). The best-known bounds on the length of a longest cycle, as of the writing of this paper, are due to Kemkes and Wormald [11], who showed that a longest cycle has typically length at most 1.739ǫ 2 n, and the very recent result of Anastos [3], who showed that a longest cycle has typically length at least 1.581ǫ 2 n. Our results imply there is a cycle of asymptotic length at least 4 3ǫ •1.1ǫ 3 n = 1.46ǫ 2 n. It is interesting to note that a careful expectation analysis of the number of paths of length k, suggests that our algorithm could construct an even longer path if one were to consider paths on only 0.98n vertices of the kernel. As one would have to prove log-concentration on the number of paths of such length, and as this would still yield a bound lower than 1.581ǫ 2 n, we did not venture in that direction.…”