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

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 $\left(\genfrac{}{}{0ex}{}{n}{3}\right)-\left(\genfrac{}{}{0ex}{}{n-m+1}{3}\right)+o(n^3)$ 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 Erdős. 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 $k,\ell ,d$.

arXiv admin note: text overlap with arXiv:0903.2867 by other authors