Berge Hamilton cycles in a random sparsification of dense hypergraphs

TL;DR

This paper extends classical hitting time results for Hamilton cycles from graphs to hypergraphs, showing that in a random process on dense hypergraphs, the emergence of Berge Hamilton cycles coincides with reaching a certain minimum degree.

Contribution

It generalizes known results by establishing hitting time thresholds for Berge Hamilton cycles in dense hypergraphs under random processes.

Findings

Hitting time for Berge Hamilton cycles matches minimum degree thresholds in hypergraphs.

Results apply to both Berge and weak Berge Hamilton cycles.

Generalizes previous work from complete hypergraphs to dense hypergraphs.

Abstract

In the standard random graph process, edges are added to an initially empty graph one by one uniformly at random. A classic result by Ajtai, Koml\'os, and Szemer\'edi, and independently by Bollob\'as, states that in the standard random graph process, with high probability, the graph becomes Hamiltonian exactly when its minimum degree becomes $2$; this is known as a \emph{hitting time} result. Johansson extended this result by showing the following: For a graph $G$ with $\delta(G) \geq (1/2+\varepsilon)n$, in the random graph process constrained to the host graph $G$, the hitting times for minimum degree $2$ and Hamiltonicity still coincide with high probability. In this paper, we extend Johansson's result to Berge Hamilton cycles in hypergraphs. We prove that if an $r$-uniform hypergraph $H$ satisfies either $\delta_1(H) \geq (\frac{1}{2^{r-1}} + \varepsilon)\binom{n-1}{r-1}$ or $\delta_2(H) \geq \varepsilon n^{r-2}$, then in the random process generated by the edges of $H$, the time at which the hypergraph reaches minimum degree $2$ coincides with the time at which it contains a Berge Hamilton cycle with high probability. In addition, we prove an analogous result for weak Berge Hamilton cycles. This generalizes the work of Bal, Berkowitz, Devlin, and Schacht, who established the result for the case where $H$ is a complete $r$-uniform hypergraph.