Transversal Hamilton cycle in hypergraph systems

Abstract

A $k$-graph system $\text{H}=\{H_i{\}}_{i\in [m]}$ 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 $\text{H}$-transversal provided that there exists an injection $\phi :E(H)\to [m]$ such that $e\in E(H_{\phi (e)})$ for all $e\in E(H)$. We show that given $k\ge 3,\gamma >0$, sufficiently large $n$ and an $n$-vertex $k$-graph system $\text{H}=\{H_i{\}}_{i\in [n]}$, if ${\delta }_{k-1}(H_i)\ge (1/2+\gamma )n$ for each $i\in [n]$, then there exists an $\text{H}$-transversal tight Hamilton cycle. This extends the result of Rödl, Ruciński and Szemerédi [Combinatorica, 2008] on single $k$-graphs.

arXiv comment20 pages,5 figures

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