# Mittagsseminar (in cooperation with A. Steger, D. Steurer and B. Sudakov)

__Mittagsseminar Talk Information__ | |

**Date and Time**: Thursday, December 09, 2021, 12:15 pm

**Duration**: 30 minutes

**Location**: OAT S15/S16/S17

**Speaker**: Kalina Petrova

## Resilience of Loose Hamilton Cycles in Random 3-Uniform Hypergraphs

Consider a random 3-uniform hypergraph H ~ H^3(np) on n vertices, where each triple of vertices form a hyperedge with probability p. In this work, we prove a resilience result for H with respect to the property of containing a loose Hamilton cycle, that is, a cycle in which consecutive edges overlap in one vertex. More specifically, we show that there is a C s.t. if p >= C n^{-3/2} log(n), H is with high probability such that any spanning subgraph of H with minimum degree at least (7/16 + o(1)) p (n-1) (n-2) / 2 has a loose Hamilton cycle. This is optimal with respect to the resilience constant, but presumably not with respect to p. We also show a corresponding result about minimum co-degree, which is optimal with respect to both the resilience constant and p. Namely, there is a C s.t. if p >= C n^{-1} log(n), H is with high probability such that any spanning subgraph of H in which each pair of vertices is in at least (1/4 + o(1)) p (n-2) edges has a loose Hamilton cycle. This is joint work with Miloš Trujić.

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