Resilience for tight Hamiltonicity

TL;DR

This paper proves that random hypergraphs are almost surely resiliently Hamiltonian, meaning they contain a tight Hamilton cycle even after certain edge removals, for various parameters.

Contribution

It establishes the resilience of Hamiltonicity in random hypergraphs with respect to subgraphs that preserve a minimum degree condition.

Findings

Random hypergraphs are almost surely resiliently Hamiltonian.

Subgraphs with minimum degree conditions still contain a tight Hamilton cycle.

The result applies to hypergraphs with parameters $k \\ge 3$ and $\gamma > 0$.

Abstract

We prove that random hypergraphs are asymptotically almost surely resiliently Hamiltonian. Specifically, for any $\gamma>0$ and $k\ge3$, we show that asymptotically almost surely, every subgraph of the binomial random $k$-uniform hypergraph $G^{(k)}\big(n,n^{\gamma-1}\big)$ in which all $(k-1)$-sets are contained in at least $\big(\tfrac12+2\gamma\big)pn$ edges has a tight Hamilton cycle. This is a cyclic ordering of the $n$ vertices such that each consecutive $k$ vertices forms an edge.