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

TL;DR

This paper determines the minimum degree thresholds for covering vertices with specific subhypergraphs in 3-uniform hypergraphs, focusing on the generalized triangle and certain trees, providing exact and asymptotic results.

Contribution

It establishes exact and asymptotic degree thresholds for covering problems involving the generalized triangle and some trees in 3-uniform hypergraphs.

Findings

Exact value of c_2(n,T) for the generalized triangle T.

Asymptotic determination of c_1(n,T).

Bounds for c_i(n,P_k) and c_i(n,S_k) for trees like paths and stars.

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 $i=1,2$.