Abstract
AbstractWe consider random subgraphs of a fixed graph with large minimum degree. We fix a positive integer k and let Gk be the random subgraph where each independently chooses k random neighbors, making kn edges in all. When the minimum degree then Gk is k‐connected w.h.p. for ; Hamiltonian for k sufficiently large. When , then Gk has a cycle of length for . By w.h.p. we mean that the probability of non‐occurrence can be bounded by a function (or ) where . © 2016 Wiley Periodicals, Inc. Random Struct. Alg., 50, 143–157, 2017
Highlights
The study of random graphs since the seminal paper of Erdos and Renyi [2] has by and large been restricted to analysing random subgraphs of the complete graph
There has been a lot of research on random subgraphs of the hypercube and grids
For 0 ≤ p ≤ 1 we let Gp be the random subgraph of G obtained by independently keeping each edge of G with probability p
Summary
The study of random graphs since the seminal paper of Erdos and Renyi [2] has by and large been restricted to analysing random subgraphs of the complete graph. There has been less research on random subgraphs of arbitrary graphs G, perhaps with some simple properties In this vain, the recent result of Krivelevich, Lee and Sudakov [8] brings a refreshing new dimension. 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 Gp 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 Gp has a cycle of length (1 − ok(1))k with probability 1 − ok(1). Krivelevich, Lee and Sudakov [9] considered a random subgraph of a “Dirac Graph” i.e. a graph with n vertices and
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