Rainbow Hamilton cycle in hypergraph system

TL;DR

This paper establishes the existence of rainbow Hamilton cycles in hypergraph systems under certain minimum degree conditions, advancing the understanding of Hamiltonicity in complex hypergraph structures.

Contribution

It introduces a new rainbow Hamilton framework for hypergraph systems and proves the existence of rainbow tight Hamilton cycles for large systems with high minimum degree.

Findings

Proves existence of rainbow tight Hamilton cycles under specified degree conditions.

Extends previous results from single graphs to hypergraph systems.

Provides a unified conclusion for all k ≥ 3.

Abstract

In this paper, we develop a new rainbow Hamilton framework, which is of independent interest, settling the problem proposed by Gupta, Hamann, M\"{u}yesser, Parczyk, and Sgueglia when $k=3$, and draw the general conclusion for any $k\geq3$ as follows. A $k$-graph system $\textbf{H}=\{H_i\}_{i\in[n]}$ is a family of not necessarily distinct $k$-graphs on the same $n$-vertex set $V$, moreover, a $k$-graph $H$ on $V$ is rainbow if $E(H)\subseteq \bigcup_{i\in[n]}E(H_i)$ and $|E(H)\cap E(H_i)|\leq1$ for $i\in[n]$. We show that given $\gamma> 0$, sufficiently large $n$ and an $n$-vertex $k$-graph system $\textbf{H}=\{H_i\}_{i\in[n]}$ , if $\delta_{k-2}(H_i)\geq(5/9+\gamma)\binom{n}{2}$ for $i\in[n]$ where $k\geq3$, then there exists a rainbow tight Hamilton cycle. This result implies the conclusion in a single graph, which was proved by Lang and Sanhueza-Matamala [$J. Lond. Math. Soc., 2022$], Polcyn, Reiher, R\"{o}dl and Sch\"{u}lke [$J. Combin. Theory \ Ser. B, 2021$] independently.