A Dirac-type Theorem for Berge Cycles in Random Hypergraphs

Abstract

A Hamilton Berge cycle of a hypergraph on $n$ vertices is an alternating sequence $(v_1, e_1, v_2, \ldots, v_n, e_n)$ of distinct vertices $v_1, \ldots, v_n$ and distinct hyperedges $e_1, \ldots, e_n$ such that $\{v_1,v_n\}\subseteq e_n$ and $\{v_i, v_{i+1}\} \subseteq e_i$ for every $i\in [n-1]$. We prove the following Dirac-type theorem about Berge cycles in the binomial random $r$-uniform hypergraph $H^{(r)}(n,p)$: for every integer $r \geq 3$, every real $\gamma>0$ and $p \geq \frac{\ln^{17r} n}{n^{r-1}}$ asymptotically almost surely, every spanning subgraph $H \subseteq H^{(r)}(n,p)$ with minimum vertex degree $\delta_1(H) \geq \left(\frac{1}{2^{r-1}} + \gamma\right) p \binom{n}{r-1}$ contains a Hamilton Berge cycle. The minimum degree condition is asymptotically tight and the bound on $p$ is optimal up to some polylogarithmic factor.