Exact Minimum Codegree Thresholds for $K_4^-$-Covering and $K_5^-$-Covering

Abstract

Given two $3$-graphs $F$ and $H$, an $F$-covering of $H$ is a collection of copies of $F$ in $H$ such that each vertex of $H$ is contained in at least one copy of them. Let $c_2(n,F)$ be the minimum integer $t$ such that every 3-graph with minimum codegree greater than $t$ has an $F$-covering.  In this note, we answer an open problem of Falgas-Ravry and Zhao (SIAM J. Discrete Math., 2016) by determining the exact value of $c_2(n, K_4^-)$ and $c_2(n, K_5^-)$, where $K_t^-$ is the complete $3$-graph on $t$ vertices with one edge removed.