Pancyclic Hamilton cycles in random graphs
Discrete Mathematics
Discrete Mathematics
Tura´n's extremal problem in random graphs: forbidding even cycles
Journal of Combinatorial Theory Series B
On the asymmetry of random regular graphs and random graphs
Random Structures & Algorithms - Special issue: Proceedings of the tenth international conference "Random structures and algorithms"
Journal of Graph Theory
A generalization of Turán's theorem
Journal of Graph Theory
Random Structures & Algorithms
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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 $\epsilon0$, 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 $\epsilon0$, 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.