The degree and codegree threshold for generalized triangle and some trees covering

Abstract

Given two $k$-uniform hypergraphs $F$ and $G$, we say that $G$ has an $F$-covering if for every vertex in $G$ there is a copy of $F$ covering it. For $1\leq i\leq k-1$, the minimum $i$-degree $\delta_i(G)$ of $G$ is the minimum integer such that every $i$ vertices are contained in at least $\delta_i(G)$ edges. Let $c_i(n,F)$ be the largest minimum $i$-degree among all $n$-vertex $k$-uniform hypergraphs that have no $F$-covering. In this paper, we consider the $F$-covering problem in $3$-uniform hypergraphs when $F$ is the generalized triangle $T$, where $T$ is a $3$-uniform hypergraph with the vertex set $\{v_1,v_2,v_3,v_4,v_5\}$ and the edge set $\{\{v_{1}v_{2}v_{3}\},\{v_{1}v_{2}v_{4}\},\{v_{3}v_{4}v_{5}\}\}$. We give the exact value of $c_2(n,T)$ and asymptotically determine $c_1(n,T)$. We also consider the $F$-covering problem in $3$-uniform hypergraphs when $F$ are some trees, such as the linear $k$-path $P_k$ and the star $S_k$. Especially, we provide bounds of $c_i(n,P_k)$ and $c_i(n,S_k)$ for $k\geq 3$, where $i=1,2$.