# Seminars & Colloquia Calendar

Discrete Math

## Powers of Hamiltonian cycles in randomly augmented graphs

#### Mathias Schacht, Yale

Location:  Hill 705
Date & time: Monday, 26 November 2018 at 2:00PM - 3:00PM

Abstract:  We study the existence of powers of Hamiltonian cycles in graphs with large minimum degree to which some additional edges have been added in a random manner. It follows from the theorems of Dirac and of Komlós, Sarközy, and Szemerédi that for every $$k$$ and sufficiently large $$n$$ already the minimum degree $$geq frac{k}{k-1}n$$ for an $$n$$-vertex graph $$G$$ alone suffices to ensure the existence of a $$k$$-th power of a Hamiltonian cycle. We show that under essentially the same degree assumption the addition of just $$O(n)$$ random edges ensures the presence of the $$(k+1)$$-st power of a Hamiltonian cycle with probability close to one.

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