Large Yk,b ${Y}_{k,b}$‐tilings and Hamilton ℓ $\ell $‐cycles in k $k$‐uniform hypergraphs

Abstract

AbstractFor , let be the ‐uniform hypergraph with two edges intersecting in vertices. Our main result is that any ‐vertex 3‐uniform hypergraph with at least edges contains a collection of vertex‐disjoint copies of , for . 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 ‐degree threshold for ‐Hamiltonicity in ‐graphs, where is odd and . Moreover, we give related results on ‐tilings and Hamilton ‐cycles with ‐degree for some other values of .