Rainbow Berge Hamiltonicity in edge-colored random $k$-uniform hypergraphs

Abstract

Let $H\sim H_c^k(n,p)$ be an edge-colored random $k$-uniform hypergraph on the vertex set $[n]$, where each edge $e\in \left(\genfrac{}{}{0ex}{}{[n]}{k}\right)$ is included independently with probability $p$ and is uniformly and independently assigned a color from the color set $[c]$. For $k=2$, Ferber and Krivelevich (2016) established that if $c=(1+o(1))n$ and $p=(\log n+\log \log n+\omega (n))/n$, then with high probability the edge-colored random graph $H\sim H_c^2(n,p)$ contains a rainbow Hamilton Berge cycle. Subsequently, Bal, Berkowitz, Devlin, and Schacht (2021) determined the threshold for the appearance of a (non-rainbow) Hamilton Berge cycle in random $k$-uniform hypergraphs. In this paper, we generalize the results to all integers $k\ge 3$. We prove that if $c=(1+o(1))n$ and $p=(k-1)!\frac{\log n+\log \log n+\omega (n)}{n^{k-1}}$, then with high probability $H\sim H_c^k(n,p)$ contains a rainbow Hamilton Berge cycle. Furthermore, both conditions on $c$ and $p$ are asymptotically tight. \noindent\emph{Key words:} Rainbow subgraph, Hamiltonicity, Berge cycle, Random hypergraph.