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- W2045530818 abstract "A graph G on n vertices is pancyclic if it contains cycles of length t for all $3leq tleq n$. 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 $pgg n^{-1+1/(l-1)}$ for some integer $lgeq3$, 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 $lleq tleq n$. 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." @default.
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- W2045530818 date "2010-01-01" @default.
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- W2045530818 title "Resilient Pancyclicity of Random and Pseudorandom Graphs" @default.
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- W2045530818 doi "https://doi.org/10.1137/090761148" @default.
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