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 maximum 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.

9 pages