Transversal Hamilton cycle in hypergraph systems

TL;DR

This paper proves the existence of transversal tight Hamilton cycles in large hypergraph systems with high minimum codegree, extending classical results from single hypergraphs to systems.

Contribution

It introduces conditions under which a hypergraph system contains a transversal tight Hamilton cycle, generalizing previous single hypergraph results to systems.

Findings

High minimum codegree ensures transversal Hamilton cycles

Extends classical hypergraph Hamilton cycle results to systems

Applicable for large hypergraph systems with specified degree conditions

Abstract

A $k$-graph system $\textbf{H}=\{H_i\}_{i\in[m]}$ is a family of not necessarily distinct $k$-graphs on the same $n$-vertex set $V$ and a $k$-graph $H$ on $V$ is said to be $\textbf{H}$-transversal provided that there exists an injection $\varphi: E(H)\rightarrow [m]$ such that $e\in E(H_{\varphi(e)})$ for all $e\in E(H)$. We show that given $k\geq3, \gamma>0$, sufficiently large $n$ and an $n$-vertex $k$-graph system $\textbf{H}=\{H_i\}_{i\in[n]}$, if $\delta_{k-1}(H_i)\geq(1/2+\gamma)n$ for each $i\in[n]$, then there exists an $\textbf{H}$-transversal tight Hamilton cycle. This extends the result of R\"{o}dl, Ruci\'{n}ski and Szemer\'{e}di [Combinatorica, 2008] on single $k$-graphs.