Long paths and cycles in random subgraphs of graphs with large minimum degree

Abstract

For a given finite graph G of minimum degree at least k, let Gp be a random subgraph of G obtained by taking each edge independently with probability p. We prove that (i) if p≥ω/k for a function ω=ω(k) that tends to infinity as k does, then Gp asymptotically almost surely contains a cycle (and thus a path) of length at least (1−o(1))k, and (ii) if p≥(1+o(1))lnk/k, then Gp asymptotically almost surely contains a path of length at least k. Our theorems extend classical results on paths and cycles in the binomial random graph, obtained by taking G to be the complete graph on k + 1 vertices. © Wiley Periodicals, Inc. Random Struct. Alg., 46, 320–345, 2015

Paths and Cycles in Random Subgraphs of Graphs with Large Minimum Degree

“…Theorem 3 improves a result of Krivelevich, Lee and Sudakov [8] and Riordan [10] implying β ′ (c) = 1 − o (1). It also generalizes several results of the length of the longest cycle in the G(n, p)-model.…”

“…Thus Theorem 2 describes precisely the asymptotic behavior of α ′ (c) as c → ∞ improving a result due to Krivelevich, Lee and Sudakov [8] who showed that α ′ (c) = 1 − O(c − 1 2 ). In addition, it generlizes results concerning the length of the longest path in the G(n, p)-model due to Ajtai, Komlós and Szemerédi [1], Fernandez de la Vega [5], Bollobás [3], Bollobás, Fenner and Frieze [4], and Frieze [6].…”

On random k‐out subgraphs of large graphs

“…In this vain, the recent result of Krivelevich, Lee and Sudakov [8] brings a refreshing new dimension. They start with an arbitrary graph G which they assume has minimum degree at least k. For 0 ≤ p ≤ 1 we let G p be the random subgraph of G obtained by independently keeping each edge of G with probability p. Their main result is that if p = ω/k then G p has a cycle of length (1 − o k (1))k with probability 1 − o k (1).…”

“…Our proofs use Depth First Search (DFS). The idea of using DFS comes from Krivelevich, Lee and Sudakov [8].…”

On the Probability that a Random Subgraph Contains a Circuit

“…Similar questions concerning cycles in random subgraphs of graphs with a given minimum degree were considered in and .…”