Large $ Y_{k,b} $-tilings and Hamilton $ \ell $-cycles in $k$-uniform hypergraphs

TL;DR

This paper establishes optimal edge thresholds for containing specific tilings and Hamilton cycles in uniform hypergraphs, advancing understanding of hypergraph matchings and Hamiltonicity.

Contribution

It proves asymptotically optimal edge bounds for $Y_{3,2}$-tilings and determines degree thresholds for Hamilton $ ext{ extlangle} ext{ extlangle} ext{ extlangle} ext{ extlangle} extlangle$-cycles in $k$-graphs, extending prior conjectures.

Findings

Optimal edge bounds for $Y_{3,2}$-tilings in 3-uniform hypergraphs.

Asymptotically best degree threshold for Hamilton $ ext{ extlangle} ext{ extlangle} extlangle$-cycles in $k$-graphs.

Results on $Y_{k,b}$-tilings and degree conditions for Hamilton cycles.

Abstract

Let $Y_{3,2}$ be the $3$-uniform hypergraph with two edges intersecting in two vertices. Our main result is that any $n$-vertex 3-uniform hypergraph with at least $\binom{n}{3} - \binom{n-m+1}{3} + o(n^3)$ edges contains a collection of $m$ vertex-disjoint copies of $Y_{3,2}$, for $m\le n/7$. The bound on the number of edges is asymptotically best possible. This problem generalizes the Matching Conjecture of Erd\H{o}s. We then use this result combined with the absorbing method to determine the asymptotically best possible minimum $(k-3)$-degree threshold for $\ell$-Hamiltonicity in $k$-graphs, where $k\ge 7$ is odd and $\ell=(k-1)/2$. Moreover, we give related results on $ Y_{k,b} $-tilings and Hamilton $ \ell $-cycles with $ d $-degree for some other values of $ k,\ell,d $.