Paths and Cycles in Random Subgraphs of Graphs with Large Minimum Degree
Abstract
For a graph G and p∈[0,1], let Gp arise from G by deleting every edge mutually independently with probability 1−p. The random graph model (Kn)p is certainly the most investigated random graph model and also known as the G(n,p)-model. We show that several results concerning the length of the longest path/cycle naturally translate to Gp if G is an arbitrary graph of minimum degree at least n−1.
For a constant c>0 and p=cn, we show that asymptotically almost surely the length of the longest path in Gp is at least (1−(1+ϵ(c))ce−c)n for some function ϵ(c)→0 as c→∞, and the length of the longest cycle is a least (1−O(c−15))n. The first result is asymptotically best-possible. This extends several known results on the length of the longest path/cycle of a random graph in the G(n,p)-model to the random graph model Gp where G is a graph of minimum degree at least n−1.