Codegree Threshold for Tiling k-graphs with Two Edges Sharing Exactly ℓ Vertices

Abstract

Given integer k and a k-graph F, let t_k-1(n, F) be the minimum integer t such that every k-graph H on n vertices with codegree at least t contains an F-factor. For integers k ≥ 3 and 0 ≤ ℓ ≤ k − 1, let ${{\cal Y}_{k,\ell }}$ be a k-graph with two edges that shares exactly ℓ vertices. Han and Zhao (J. Combin. Theory Ser. A, (2015)) asked the following question: For all k ≥ 3, 0 ≤ ℓ ≤ k − 1 and sufficiently large n divisible by 2k − ℓ, determine the exact value of $t_{k-1}\left( {n,\;{{\cal Y}_{k,\ell }}} \right)$. In this paper, we show that $t_{k-1}\left( {n,\;{{\cal Y}_{k,\ell }}} \right) = {n \over {2k - \ell }}$ for k ≥ 3 and 1 ≤ ℓ ≤ k − 2, combining with two previously known results of Rödl, Ruciński and Szemerédi (J. Combin. Theory Ser. A, (2009)) and Gao, Han and Zhao (Combinatorics, Probability and Computing, (2019)), the question of Han and Zhao is solved completely.