Resilience for Loose Hamilton Cycles

TL;DR

This paper investigates the conditions under which loose Hamilton cycles appear in random hypergraphs, establishing thresholds that match dense hypergraph cases for certain probabilities and degree conditions.

Contribution

It determines the minimum degree threshold for loose Hamiltonicity in subgraphs of random hypergraphs, linking it to dense hypergraph thresholds for specific probability ranges.

Findings

Threshold for loose Hamilton cycles matches dense case when p ≥ n^{-(k-1)/2+o(1)}.

Approximate tightness of p for degrees d > (k+1)/2.

Dense threshold unknown beyond degrees d ≥ k-2.

Abstract

We study the emergence of loose Hamilton cycles in subgraphs of random hypergraphs. Our main result states that the minimum $d$-degree threshold for loose Hamiltonicity relative to the random $k$-uniform hypergraph $H_k(n,p)$ coincides with its dense analogue whenever $p \geq n^{- (k-1)/2+o(1)}$. The value of $p$ is approximately tight for $d>(k+1)/2$. This is particularly interesting because the dense threshold itself is not known beyond the cases when $d \geq k-2$.