## Choongbum Lee (이중범), Resilient pancyclicity of random graphs

Resilient pancyclicity of random graphs
Choongbum Lee (이중범)
Department of Mathematics, UCLA, Los Angeles, USA
2009/7/31 Thursday 4PM-5PM

A graph G on n vertices is pancyclic if it contains cycles of length t for all $$3 \leq t \leq n$$. 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.

Joint work with Michael Krivelevich and Benny Sudakov

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