Exact minimum codegree thresholds for $K_4^-$-covering and $K_5^-$-covering

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.


Introduction
Given a set V and a positive integer k, let V k be the collection of k-element subsets of V . A simple k-uniform hypergraph (or k-graph for short) H = (V, E) consists of a vertex set V and an edge set E ⊆ V k . We write graph for 2-graph for short. For a set S ⊆ V (H), the neighbourhood N H (S) of S is {T ⊆ V (H)\S : T ∪ S ∈ E(H)} and the degree of S is d H (S) = |N H (S)|. The minimum (resp. maximum) s-degree of H, denoted by δ s (H) (resp. ∆ s (H)), is the minimum (resp. maximum ) d H (S) taken over all s-element sets of V (H). δ k−1 (H) and δ 1 (H) are usually called the minimum codegree and the minimum degree of H, respectively. An r-graph H is called an r-partite r-graph if the vertex set of H can be partitioned into r parts such that each edge of H intersects each part exactly one vertex. Given disjoint sets V 1 , V 2 , · · · , V r , let K(V 1 , V 2 , . . . , V r ) be the complete r-partite r-graph with vertex classes V 1 , V 2 , . . . , V r .
Given a k-graph F , we say a k-graph H has an F -covering if each vertex of H is contained in some copy of F . For 0 i < k, define c i (n, F ) = max{δ i (H) : H is a k-graph on n vertices with no F -covering}.
We call c k−1 (n, F ) the minimum codegree threshold for F -covering.
There are two well studied extremal problems related to the covering problem. Given a k-graph F , a k-graph H is F -free if H does not contain a copy of F as a subgraph. For 0 i < k, define The quantity ex 0 (n, F ) is known as the Turán number of F , and ex k−1 (n, F ) was studied by Mubayi and Zhao [7]. For an overview of the Turán problem for hypergraphs, one can see a survey given by Keevash [5]. Given two k-graphs H and F with |V (H)| is divisible by |V (F )|, a perfect F -tiling (or an F -factor) in H is a spanning collection of vertex-disjoint copies of F . For 0 i < k and n divisible by |V (F )|, define The study of the tiling problem also has a long history. For detailed discussion of the area, one can refer to the surveys due to Rödl and Ruciński [8] and Zhao [10]. Trivially, for 0 i < k, we have ex i (n, F ) c i (n, F ) t i (n, F ).
So the covering problem is an intermediate but distinct problem from the well-studied Turán and tiling problems. This is also partial motivation for the study of covering problems. For graphs F , the F -covering problem was solved asymptotically in [9] by showing that c 1 (n, F ) = ( χ(F )−2 χ(F )−1 + o(1))n, where χ(F ) is the chromatic number of F . For general kgraphs, the function c i (n, F ) was determined for some special families of k-graphs F . For example, Han, Lo, and Sanhueza-Matamala [3] proved that c k−1 (n, C (k,k−1) s ) ( 1 2 +o(1))n for k 3, s 2k 2 and the result is asymptotically tight if k and s satisfy some special constrains, where C (k, ) s (1 < k) is the k-graph on s vertices such that its vertices can be ordered cyclicly so that every edge consists of k consecutive vertices under this order and two consecutive edges intersect in exactly vertices. Han, Zang, and Zhao showed in [4] that c 1 (n, K) = (6−4 √ 2+o(1)) n 2 , where K is a complete 3-partite 3-graph with at least two vertices in each part. Let K t denote the complete 3-graph on t vertices and let K − t denote the 3-graph obtained from K t by removing one edge. Recently, Falgas-Ravry, Markström, and Zhao [1] asymptotically determined c 1 (n, K − 4 ) and gave close to optimal bounds for c 1 (n, K − 4 ). In this note, we focus on the problem to determine the exact value of c 2 (n, K − t ) when t = 4 and 5. Falgas-Ravry and Zhao [2] determined the exact value of c 2 (n, K 4 ) for n > 98 and gave lower and upper bounds of c 2 (n, K − 4 ) and c 2 (n, K − 5 ). More specifically, they proved the following theorem. Theorem 1.1 (Theorem 1.2 in [2]). Suppose n = 6m + r for some r ∈ {0, 1, 2, 3, 4, 5} and m ∈ N with n 7. Then Falgas-Ravry and Zhao [2] also conjectured that the gap between the upper and lower bounds for c 2 (n, K − 4 ) could be closed and left this as an open problem.
In this note, we determine not only the exact value of c 2 (n, K − 4 ) but also the exact value of c 2 (n, K − 5 ), thereby resolving Problem 1.3 and sharpening Theorem 1.5.
The following are some definitions and notation used in our proofs. For a k-graph H and x ∈ V (H), the link graph of x, denoted by H(x), is the (k − 1)-graph with vertex set V (H) \ {x} and edge set N H (x). Given a graph G and a positive integer vector k ∈ Z V (G) + , the k-blowup of G, denoted by G (k) , is the graph obtained by replacing every vertex v of G with an independent k(v)-set X v , and placing a complete bipartite graph between X u and X v whenever u and v are adjacent in G. We call the independent set X v in G (k) the blowup of v in G. When there is no confusion, we write ab and abc as a shorthand for {a, b} and {a, b, c}, respectively. Given a positive integer n, write [n] for the set {1, 2, . . . , n}.
In the rest of the note, we give proofs of Theorems 1.4 and 1.5.
2 Proof of Theorems 1.4 and 1.5 We will construct extremal 3-graphs for K − 4 and K − 5 with minimum codegree matching the upper bounds in Theorems 1.1 and 1.2, respectively.

Proof of Theorem 1.4
We first give an observation, which can be verified directly from the definitions. Observation 1. Let H be a 3-graph and x ∈ V (H). x is not covered by a copy of K − 4 if and only if (i) H(x) is triangle-free, and (ii) every edge in H induces at most one edge in H(x).
By Theorem 1.1, to show Theorem 1.4, it is sufficient to construct 3-graphs H on n vertices for n ≡ 0, 3, 4 (mod 6) and with δ 2 (H) = n 3 such that H has no K − 4 -covering. In the proof we distinguish three cases. Let C 6 be the 6-cycle v 1 v 2 v 3 v 4 v 5 v 6 v 1 and let 12 . . . k1 be a k-cycle on vertices 1, 2, . . . , k for some positive integer k.
It can be checked that G 1 , G 2 , G 3 are triangle-free (see Fig.1); therefore, so are their blowups.   N H 1 (x, a)∩N H 1 (a, b) Case 1 follows directly from Claim 1.
is triangle-free, too; and by (2) of Construction 2, any two incident edges of H 2 (x) are not contained in one edge of H 2 . By Observation 1, x is contained in no copy of K − Case 2 follows from Claim 2.

Proof of Theorem 1.5
The following theorem is well known in graph theory.
Theorem 2.1 (König [6]). Let G be a bipartite graph with maximum degree ∆. Then E(G) can be partitioned into M 1 , M 2 , . . . , M ∆ so that each M i (1 i ∆) is a matching in G.
In particular, if G is ∆-regular then E(G) can be partitioned into ∆ perfect matchings. Proof of Theorem 1.5. We first give the extremal 3-graph for K − 5 . Construction 5. Given a positive integer m and three disjoint sets Let T be the 3-partite 3-graph on vertex set V 1 ∪ V 2 ∪ V 3 constructed by Construction 4. Let x be a specific vertex not belonging to V 1 ∪ V 2 ∪ V 3 . Define the 3-graph H 4 on vertex set V 1 ∪ V 2 ∪ V 3 ∪ {x} such that the following holds.
(1) The link graph of x, H 4 (x), consists of the union of the three complete bipartite Remark. By the definition of Construction 5, we have 3m − 1  .

Proof of Claim 4:
We show that x is contained in no copy of K − 5 in H 4 . Choose a 4-set {a, b, c, d} ⊆ V 1 ∪ V 2 ∪ V 3 . If it contains at least three vertices in the same part V i or at least two vertices in at least two different parts V i , V j , then by (1) of Construction 5, {a, b, c, d} spans at least two non-edges in H 4 (x). Otherwise, two vertices in {a, b, c, d}, say a, b, lie in the same part (whence they span a non-edge in H 4 (x)) while the other two vertices c and d lie one each in the two other parts. Since ∆ 2 (T ) 1 (by Construction 4) and (3) of Construction 5, at most one of a and b makes an edge of T (and hence H 4 ) with cd. Thus in either case, {x, a, b, c, d} spans at least two non-edges of H 4 and hence x is not covered by a copy of K − 5 . Now we compute the minimum codegree of H 4 . Choose two distinct vertices a, b ∈ V (H 4 ). If x ∈ {a, b}, assume x = a and b ∈ V i , then by (1) of Construction 5, If a, b ∈ V i for some 1 By Theorem 1.2, we have Theorem 1.5.