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

  • Stefan Ehard
  • Felix Joos
Keywords: Graph Theory, Random graphs, Cycles, Paths, Minimum Degree

Abstract

For a graph G and p[0,1], let Gp arise from G by deleting every edge mutually independently with probability 1p. 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 n1.

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))cec)n for some function ϵ(c)0 as c, and the length of the longest cycle is a least (1O(c15))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 n1.

Published
2018-05-25
Article Number
P2.31