Resilient pancyclicity of random and pseudo-random graphs

TL;DR

This paper proves that random and pseudo-random graphs with certain edge probabilities are resiliently pancyclic, maintaining cycles of all lengths despite some edge removals, with tight bounds on parameters.

Contribution

It establishes tight resilience thresholds for pancyclicity in random and pseudo-random graphs under edge deletions.

Findings

Random graphs with p(n) >> n^{-1/2} are resiliently pancyclic.

Subgraphs with degree constraints preserve cycles of all lengths.

Results are tight with respect to probability bounds and degree constants.

Abstract

A graph $G$ on $n$ vertices is \textit{pancyclic} if it contains cycles of length $t$ for all $3 \leq t \leq n$. In this paper we prove that for any fixed $\epsilon>0$, the random graph $G(n,p)$ with $p(n)\gg n^{-1/2}$ asymptotically almost surely has the following resilience property. If $H$ is a subgraph of $G$ with maximum degree at most $(1/2 - \epsilon)np$ then $G-H$ is pancyclic. In fact, we prove a more general result which says that if $p \gg n^{-1+1/(l-1)}$ for some integer $l \geq 3$ then for any $\epsilon>0$, asymptotically almost surely every subgraph of $G(n,p)$ with minimum degree greater than $(1/2+\epsilon)np$ contains cycles of length $t$ for all $l \leq t \leq n$. These results are tight in two ways. First, the condition on $p$ essentially cannot be relaxed. Second, it is impossible to improve the constant 1/2 in the assumption for the minimum degree. We also prove corresponding results for pseudo-random graphs.