Resilient Pancyclicity of Random and Pseudorandom Graphs

Abstract

A graph G on n vertices is 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 $p(n)\gg n^{-1/2}$ (i.e., with $p(n)/n^{-1/2}$ tending to infinity) 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 $l\geq3$, 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 pseudorandom graphs.