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

TL;DR

This paper extends known results about the longest paths and cycles in random graphs to a broader class of graphs with large minimum degree, providing asymptotically optimal bounds for these lengths.

Contribution

It generalizes classical results from the Erdős–Rényi model to arbitrary graphs with high minimum degree, establishing new bounds on longest paths and cycles in their random subgraphs.

Findings

Longest path length at least (1-(1+ε(c))ce^{-c})n asymptotically almost surely

Longest cycle length at least (1-O(c^{-1/5}))n asymptotically almost surely

Results extend known properties of G(n,p) to graphs with large minimum degree

Abstract

For a graph $G$ and $p\in [0,1]$, let $G_p$ arise from $G$ by deleting every edge mutually independently with probability $1-p$. The random graph model $(K_n)_p$ 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 $G_p$ if $G$ is an arbitrary graph of minimum degree at least $n-1$. For a constant $c$, we show that asymptotically almost surely the length of the longest path is at least $(1-(1+\epsilon(c))ce^{-c})n$ for some function $\epsilon(c)\to 0$ as $c\to \infty$, and the length of the longest cycle is a least $(1-O(c^{- \frac{1}{5}}))n$. The first result is asymptotically best-possible. This extents several known results on the length of the longest path/cycle of a random graph in the $G(n,p)$-model.