Exact minimum codegree thresholds for \(K_4^-\)-covering and \(K_5^-\)-covering (Q785570)

Summary: 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 \textit{V. Falgas-Ravry} and \textit{Y. Zhao} [SIAM J. Discrete Math. 30, No. 4, 1899--1917 (2016; Zbl 1346.05199)] 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.