Codegree threshold for tiling balanced complete 3-partite 3-graphs and generalized 4-cycles

Given two k-graphs F and H, a perfect F -tiling (also called an F -factor) in H is a set of vertex-disjoint copies of F that together cover the vertex set of H. Let tk−1(n, F ) be the smallest integer t such that every k-graph H on n vertices with minimum codegree at least t contains a perfect F -tiling. Mycroft (JCTA, 2016) determined the asymptotic values of tk−1(n, F ) for k-partite k-graphs F and conjectured that the error terms o(n) in tk−1(n, F ) can be replaced by a constant that depends only on F . In this paper, we determine the exact value of t2(n,K 3 m,m), where K3 m,m (defined by Mubayi and Verstraëte, JCTA, 2004) is the 3-graph obtained from the complete bipartite graph Km,m by replacing each vertex in one part by a 2-elements set. Note that K3 2,2 is the well known generalized 4-cycle C 3 4 (the 3-graph on six vertices and four distinct edges A,B,C,D with A ∪B = C ∪D and A∩B = C∩D = ∅). The result confirms Mycroft’s conjecture for K3 m,m. Moreover, we improve the error term o(n) to a sub-linear term when F = K3(m) and show that the sub-linear term is tight for K3(2), where K3(m) is the complete 3-partite 3-graph with each part of size m. Mathematics Subject Classifications: 05C35, 05C65,05C70 ∗Supported by National Nature Science Foundation of China (No. 11671376) and Anhui Initiative in Quantum Information Technologies (AHY150200). the electronic journal of combinatorics 27(3) (2020), #P3.47 https://doi.org/10.37236/9061


Introduction
A k-graph H is a pair H = (V, E) where V is a set of elements called vertices, and E is a collection of subsets of V with uniform size k called edges. We call |V | the order of H and |E| the size of H, also denoted by |H| or e(H). We write graph for 2-graph for short. Given two k-graphs F and H, an F -tiling in H is a collection of vertex-disjoint copies of F in H. An F -tiling is perfect if it covers every vertex of H, also known as an F -factor. If F is a single edge then an F -factor in H is a perfect matching in H. As for matchings, a natural question for tiling is to determine the minimum degree threshold for finding a perfect F -tiling. Given S ⊆ V (H), the degree of S, denote by d H (S), is the number of edges of H containing S. The minimum s-degree δ s (H) of H is the minimum of d H (S) over all S ⊆ V (H) of size s. For integer n divisible by |V (F )|, define t s (n, F ) to be the smallest integer t such that every k-graph H on n vertices with δ s (H) t contains a perfect F -tiling. For n ∈ N, write [n] for the set {1, . . . , n}, and rN for the set of positive integers divisible by integer r Tiling problems have been widely studied for graphs. The celebrated Hajnal-Szemerédi Theorem [8] states that t 1 (n, K r ) = (1−1/r)n for n ∈ rN. Alon and Yuster [1] generalized the Hajnal-Szemerédi Theorem to t 1 (n, H) (1 − 1/χ(H))n + o(n) for every H with chromatic number χ(H) and n ∈ hN; later, Komlós, Sárközy, and Szemerédi [15] proved that the error term o(n) can be replaced by a constant C = C(H). In [19], Kühn and Osthus improved Alon-Yuster's result to t 1 (n, H) = (1 − 1/χ * (H))n + O(1), where χ * (H) depends on the relative sizes of the colour classes in the optimal colourings of H and satisfies χ(H) − 1 χ * (H) χ(H). See [18] for a survey on graph tiling.
For hypergraphs, we know much less and tiling problems become much harder. There are a number of research results on perfect matching problem, see [26,28] for surveys.
A (k, )-cycle C (k, ) s is a k-graph on s vertices so that whose vertices can be ordered cyclically in such a way that the edges are sets of consecutive k vertices and every two consecutive edges share exactly vertices. Gao and Han [6] and Czygrinow [2] determined the exact value of t 2 (n, C (3,1) 6 ) and t 2 (n, C (3,1) s )(s 6), respectively, and Gao, Han and Zhao [7] determined t k−1 (n, C (k,1) s ) for k 4. Han, Lo, and Sanhueza-Matamala [11] proved t k−1 (n, C (k,k−1) s ) (1/2 + 1/(2s) + o(1))n where k 3 and s 5k 2 and this bound is asymptotically best possible for infinitely many pairs of s and k.
In the study of tiling problems, another family of hypergraphs which was well studied are k-partite k-graphs. A k-graph F on vertex set V is said to be k-partite if V can be partitioned into vertex classes V 1 , . . . , V k so that for any e ∈ F and 1 j k we have |e ∩ V j | =1. The partition V 1 , . . . , V k of V is called a k-partite realisation of V . Define where in each case the union is taken over all k-partite realisations The greatest common divisor of F , denoted by gcd(F ), is then defined to be the greatest common divisor of the set D(F ) (if D(F ) = {0} then gcd(F )is undefined). The smallest class ratio of F , denoted by σ(F ), is defined by Note in particular that σ(F ) 1/k, with equality if and only if Observe that a complete k-partite k-graph has only one k-partite realisation up to permutations of the vertex classes V 1 , . . . , V k . Hence, we write [23] proved a general result on tiling k-partite k-graphs.  [23]). Let F be a k-partite k-graph. Then for any α > 0 there exists n 0 such that if H is a k-graph on n n 0 vertices for which |V (F )| divides n and if gcd(F ) = 1; max{σ(F )n, n p } + αn if gcd(S(F )) = 1 and gcd(F ) > 1, then H contains a perfect F -tiling, where p is the smallest prime factor of gcd(F ). Moreover, (1) is asymptotically best possible for a large class of k-partite k-graphs including complete k-partite k-graphs.
Furthermore, Mycroft also conjectured that the error terms in (1) can be replaced by a (sufficiently large) constant that depends only on F . 23]). Let F be a k-partite k-graph. Then there exists a constant C = C(F ) such that the error terms in (1) can be replaced by C.
Gao, Han and Zhao [7] improved the error term for complete k-partite k-graphs F = K k (a 1 , . . . , a k−1 , a k ) with gcd(F ) = 1 and disproved Conjecture 1.2 for all complete kpartite k-graphs F with gcd(F ) = 1 and a k−1 2 (remark: in the updated version of [7], the authors constructed more counterexamples for the conjecture of Mycroft). Han, Zang, and Zhao [13] determined t 1 (n, K) asymptotically for all complete 3-partite 3-graphs K. In this paper, we focus on balanced complete 3-partite 3-graphs. One of our main results is the following. Theorem 1.3. Let m 2 be an integer. There exists an integer n 0 ∈ N such that the following holds. Suppose that H is a 3-graph on n n 0 vertices with n ∈ 3mN. If δ 2 (H) n/2 + m 1 m n 1− 1 m then H contains a K 3 (m)-factor. For K 3 (2), we show that the lower bound of δ 2 (H) is tight up to a factor. Proposition 1. There exists an integer n 1 ∈ N. For every n n 1 , there exists a 3-graph H on n vertices with δ 2 (H) n/2 + √ 2n/5 − 3 containing no K 3 (2)-factor.
Clearly, Theorem 1.3 improves the error term αn in (1) to Cn 1−1/m when F = K 3 (m), and Proposition 1 shows that the error term C √ n can not be replaced by a constant for F = K 3 (2) and henceforth for F = K 3 (2m), which gives a new family of counterexamples for Conjecture 1.2 (As mentioned in the end of [7], K 3 (2) is not included in the family of counterexamples given by Gao, Han and Zhao).
Given integer k, let C k 4 be the family of k-graphs which contains four distinct edges A, B, C, D with A ∪ B = C ∪ D and A ∩ B = C ∩ D = ∅, which was first introduced by Erdős [4], and is also called the generalized 4-cycles. For k = 2 or 3, we write C k 4 for C k 4 instead because there is only one graph, up to isomorphism, in C k 4 in these cases. Note that C 3 4 is a supported subgraph of K 3 (2). Let X 1 , X 2 , . . . , X t be t pairwise disjoint sets of size k − 1 and let Y be a set of s In [25], Mubayi and Verstraëte investigated the Turán number of K 3 s,t . We show that Conjecture 1.2 is valid for K 3 m,m , in particular for generalized 4-cycle since K 3 2,2 = C 3 4 . More precisely, we prove the following theorem. Theorem 1.4. For any integer m, there exists an integer N such that for all n ∈ 3mN and n N , To show the lower bound of t 2 (n, K 3 m,m ) in Theorem 1.4 is tight, we give a construction of extremal 3-graph for K 3 m,m . In the following we give some notation used in this paper. For a k-graph H = (V, E) and a vertex set U ⊆ V , write H[U ] for the subgraph of H induced by U and U r for the set of all subsets of size r of U . For an S ⊆ V , the neighbourhood of S, denoted by N H (S) or N (S) if there is no confusion from the context, is the set of subsets T ⊆ V such that S ∪ T ∈ E(H), the link graph of S, denoted by H S , is the (k − |S|)-graph with vertex set V (H) \ S and edge set N H (S). For a 3-graph H = (V, E) and u, v, w ∈ V , we write uv and uvw for the sets {u, v} and {u, v, w}, respectively. Let V 1 , . . . , V t be a partition of . Given two constants α and β, we write α β if α is sufficiently small with respect to β.

Lemmas and proofs of main results
To show Proposition 1, we first revisit a construction of K k (1, . . . , 1, 2, t + 1)-free (t 1) k-graph G with e(G) ∼ √ t k! n k− 1 2 edges given by Mubayi [24]. We only need the special case that k = 3 and t = 1. Let q be a prime power and F q be the q-element finite field.

As shown in
For convenience, we use ordered triple (a, b, c) denote an edge of H q with a, b ∈ V (G q ) and c ∈ V (G q ).
Remark. By the constructions of G q and H q , we know that an edge e = abc ∈ E(G q ) corresponds to three edges e 1 = (a, b, c), e 2 = (a, c, b), e 3 = (b, c, a) in H q , and H q possibly contains some edges of the form (a, b, a) or (a, b, b). The following fact shows that H q inherits some properties from G q .
Proof. We show that H q is also K 3 (1, 2, 2)-free. As shown in [24], for ( has at most one solution (x, y) if (p 1 , p 1 ) = (p 2 , p 2 ). Suppose that H q contains a copy of the electronic journal of combinatorics 27(3) (2020), #P3.47 (v 2 , v 2 ). Without loss of generality, we may assume a, b 1 . So the equation system (3) has at most one solution, this is a contradiction to Proof of Proposition 1: For sufficiently large n, without loss of generality, we may assume n ∈ 6N, choose an odd prime power q and n 0 = (q − 1) 2 such that n/2 + 2 5 n/2 n 0 n/2 + 1 2 n/2. Let F q be the q-element finite field and let A and B be the sets obtained by deleting any one element and 2n We claim that H does not contain a K 3 (2)-factor. Suppose to the contrary that H contains a K 3 (2)-factor. Since |A| is odd, H must contain a copy of K 3 (2) such that |V (K 3 (2)) ∩ A| is odd. Such a copy of K 3 (2) must be of type (5, 1) or (3,3). Note that copies of K 3 (2) in B[A, B] must intersect A in an even number of vertices. It is an easy task to check that a copy of K 3 (2) of type (5, 1) or (3, 3) forces a copy of K 3 (1, 2, 2) in H , a contradiction.
The proof of Theorems 1.3 and 1.4 are separated into non-extremal case and extremal case. For the non-extremal case, we use the standard absorbing method, which has been introduced by Rödl, Ruciński and Szemerédi in [27] and widely used in different research papers for example in [3,13,21].
Roughly speaking, our proof follows two steps: first, we use an "absorbing lemma" to find a small absorbing set W ⊂ V (H) with the property that given any "sufficiently small" set U ⊂ V (H) \ W , both H[W ] and H[W ∪ U ] contain K 3 (m)-factors; second, we use an"almost tiling lemma" to find a K 3 (m)-tiling in H \ W that covers all but at most o(n) vertices. The first step will be completed in Lemma 2.1 and the second step has been done by an almost tiling lemma given by Mycroft in [23], we restate it in Lemma 2.2.
Given γ > 0, H and G are two 3-graphs on the same vertex set V . We say that 1 and m be an positive integer. Suppose that H is a 3-graph of order n with δ 2 (H) (1/2 − γ)n. If H is not 3γ-extremal, then there exists a set W ⊂ V (H) with |W | 1 n and |W | ∈ 3mN, so that for any Lemma 2.2 (Almost tiling lemma, Lemma 1.5 in [23]). Let K be a k-partite k-graph. Then there exists a constant C = C(K) such that for any α > 0 there exists an integer n 0 = n 0 (K, α) with the property that every k-graph H on n n 0 vertices with δ k−1 (H) (σ(K) + α)n admits a K-tiling covering all but at most C vertices of H. Lemmas 2.3 and 2.4 deal with the extremal case for K 3 (m) and K 3 m,m , respectively. Lemma 2.3. Let m 2 be an integer. There exist γ > 0 and n 0 ∈ N such that the following holds. Suppose that H is a 3-graph on n n 0 vertices with δ 2 (H) Lemma 2.4. There exist γ > 0 and n 0 ∈ N such that the following holds. Suppose that H is a 3-graph on n n 0 vertices with δ 2 (H) satisfying (2), where n ∈ 3mN. If H is γ-extremal, then H contains a K 3 m,m -factor.
Proof of Theorems 1.3 and 1.4: Let 0 < α 1 and 1/n (2) The rest of the paper is organized as follows. In Section 3, we give the proof of the absorption lemma used in the paper, i.e. Lemma 2.1, and in Section 4, we deal with the extremal case, i.e. we prove Lemmas 2.3 and 2.4.

Absorption lemma
To prove the absorption lemma, we need some preliminaries. Let H = (V, E) be a k-graph of order n, and F be a k-graph of order t. Given an integer i 1, a constant η > 0, and A subset U ⊂ V is said to be (F, i, η)-closed in H if every pair of vertices in U are (i, η)-close with respect to F . If V is (F, i, η)-closed in H then we simply say that H is (F, i, η)-closed.
The following lemma given by Lo and Markström [21] referred to as absorption lemma provides an absorbing set for any sufficiently small vertex set if H is (F, i, η)-closed. Lemma 3.1 (Lemma 1.1 in [21]). Let t and i be positive integers and η > 0. Then there exist η 1 , η 2 such that 0 < η 2 η 1 η and an integer n 0 = n 0 (i, η) satisfying the following: Suppose that F is a k-graph of order t and H is an (F, i, η)-closed k-graph of order n n 0 . Then there exists a vertex subset U ⊂ V (H) of size at most η 1 n with |U | ∈ tZ such that, for every vertex set W ⊂ V \ U of size at most η 2 n with |W | ∈ tZ, both H[U ] and H[U ∪ W ] contain F -factors.
Lemma 3.2 also given in [21] allows us to find close pairs with respect to a k-partite k-graph F . Lemma 3.2 (Lemma 4.2 in [21]). Let k 2 be an integer and α > 0. Given a k-partite k-graph F , there exist a constant η 0 = η 0 (k, F, α) > 0 and an integer n 0 = n 0 (k, F, α) such that the following holds: Let H be a k-graph of order n n 0 and x, y ∈ V (H). If then x and y are (F, 1, η)-close for all 0 < η η 0 .
The following lemma in [12] gives us a partition of V (H) with bounded number of parts such that each of them is closed with respect to F . Lemma 3.3 (Lemma 6.3 in [12]). Given δ > 0, integers c, k, t 2 and 0 < η 1/c, δ, 1/t, there exists a constant η > 0 such that the following holds for all sufficiently large n: Let F be a k-graph on t vertices. Assume a k-graph H on n vertices satisfies that |Ñ F,1,η (v)| δn for any v ∈ V (H) and every set of c + 1 vertices in V (H) contains two vertices that are (F, 1, η)-close. Then we can find a partition of V (H) into V 1 , . . . , V r with r min{c, 1/δ} such that for any j ∈ [r], |V j | (δ − η)n and V j is (F, 2 c−1 , η )-closed in H.
Actually here we use a variant absorbing method which is so-called lattice-based absorption developed by Han [9], the notation used were first given by Keevash and Mycroft [14]. Given a k-graph H = (V, E) and a partition P = {V 1 , . . . , V r } of V , the index vector i P (S) of a subset S ⊂ V with respect to P is the vector whose j-th coordinate is the size of the intersection of S and V j . A vector v ∈ Z r is called an s-vector if all its coordinates are nonnegative and their sum equals to s. Given a k-graph F of order t and µ > 0, a t-vector v is called a µ-robust F -vector if there are at least µn t copies F of F in H satisfying i P (V (F ))=v. Let I µ P,F (H) be the set of all µ-robust F -vectors and L µ P,F (H) be the lattice (i.e. the additive subgroup) generated by I µ P,F (H). For j ∈ [r], let u j ∈ Z r be the j-th unit vector, namely, u j has 1 on the j-th coordinate and 0 on other coordinates. A transferral is a vector of the form u j − u for some distinct j, ∈ [r].
The following lemma in [13] states that if L µ P,F (H) contains all transferrals then H is closed.
the electronic journal of combinatorics 27(3) (2020), #P3.47 Lemma 3.4 (Lemma 3.9 in [13]). Let i 0 , k, r 0 > 0 be integers and let F be a k-graph on t vertices. Given constants , η, µ > 0, there exist η > 0 and an integer i 0 0 such that the following holds for sufficiently large n: Let H be a k-graph on n vertices with a partition P = {V 1 , . . . , V r } such that r r 0 and for each j ∈ [r], |V j | n and V j is (F, i 0 , η)-closed in H. If u j − u ∈ L µ P,F (H) for all 1 j < r, then H is (F, i 0 , η )-closed.
The following lemma helps us to count the number of copies of K 3 (m).
Lemma 3.5 (Corollary 2 in [5]). Let F be a k-partite k-graph of order t. For every > 0, there exists a constant µ > 0 and an integer n 0 such that every k-graph H of order n n 0 with e(H) > n k contains at least µn t copies of F .
We also need the following lemma from [10].
Lemma 3.6 (Lemma 3.3 in [10]). Let 0 < 1/n γ < 1/100. Suppose that H is a 3graph of order n with δ 2 (H) (1/2 − γ)n. Let X, Y be any bipartition of V (H) with |X|, |Y | n/5. If H is not 3γ-extremal, then H contains at least γ 2 n 3 XXY -edges and at least γ 2 n 3 XY Y -edges. Now it is ready to give the proof of our absorption lemma, we restate it here. Proof. Assume γ is sufficiently small and let α = γ/3. Let F = K 3 (m). If we prove that H is (F, i, η)-closed for some i > 0 and 0 < η γ, then by Lemma 3.1 with t = 3m we obtain the desired absorbing set. So in the following it is sufficient to show that H is (F, i, η)-closed for some parameters i > 0 and 0 < η γ. The outline of the proof is as follows. The first step is, by applying Lemma 3.3 on H, to obtain a partition P of V (H) with |P| 2 such that each part is (F, 2, η )-closed and has large enough size. To show that all conditions of Lemma 3.3 are satisfied, we need to verify that for every vertex v ∈ V (H),Ñ F,1,η (v) is large enough (this can be done in Claim 2) and any three vertices contain at least one (F, 1, η)-close pair (this can be done by using Lemma 3.2). If |P| = 1, then we are done. Otherwise, we show H is closed by applying Lemma 3.4 on H and P, i.e. we prove that L µ P,F (H) contains all transferrals (this can be done in Claims 3 and 4). Claim 2. For each v ∈ V (H) and some 0 < η γ,Ñ F,1,η (v) (1/2 − 2γ)n.

Proof of Claim 2:
Fix v ∈ V (H), we have Together with |N (S)| (1/2 − γ)n, we have Given any three vertices x 1 , x 2 , x 3 ∈ V (H), by (4) and the inclusion-exclusion principle, we have By the pigeonhole principle, there exists at least one pair x i , x j so that |N (x i ) ∩ N (x j )| α n 2 , by Lemma 3.2, such a pair x i , x j is (F, 1, η)-close. Now apply Lemma 3.3 to F and H with δ = (1/2 − 2γ), c = 2 and η γ, we have that there exist a constant η > 0 and a partition P of V with at most 2 parts such that each part has size at least (1/2 − 3γ)n and is (F, 2, η )-closed in H. If |P| = 1, then H is (F, 2, η )-closed, as desired. So, we assume |P| = 2 and P = {X, Y }. Since H is not 3γ-extremal, by Lemma 3.6, both e(XXY ) and e(XY Y ) are at least γ 2 n 3 .
Define E 0 = {xy : x ∈ X, y ∈ Y, d X (xy) γ 2 n, d Y (xy) γ 2 n}, Lemma 3.4, to show that H is closed it suffices to show u 1 −u 2 ∈ L µ P,F (H) for some µ. Or equivalently, we need to show that there exists an such that H contains at least µn 3m copies of K 3 (m) of types ( , 3m − ) and ( + 1, 3m − − 1), respectively. We split the following proof into two cases according to the size of E 0 . Claim 3. There exists µ 1 > 0 for any given integers 0 s, t m with s + t = m such that the following holds: If |E 0 | γ 4 n 2 , then H contains at least µ 1 n 3m copies of K 3 (m) of type (m + s, m + t).
Proof of Claim 3: Choose 0 < γ 1 γ. Construct an auxiliary 4-partite 4-graph G 1 as follows. Let V (G 1 ) = X ∪ X ∪ Y ∪ Y , where X and Y are copies of X and Y , respectively; for a ∈ X , x ∈ X, y ∈ Y, b ∈ Y , axyb ∈ E(G 1 ) if and only if axy, xyb ∈ H. Then |V (G 1 )| = 2n, and Hence, by Lemma 3.5, there exists a positive constant µ 1 such that G 1 contains at least µ 1 n 4m copies of K 4 (m) . Fix a pair (s, t), a copy of K 4 (s, m, m, t) is contained in at most m, m, m). Therefore, G 1 contains at least µ 1 n 3m copies of K 4 (s, m, m, t) for some µ 1 > 0. Observe that a copy of K 4 (s, m, m, t) in G 1 gives us a copy of K 3 (m) of type (m + s, m + t). Then H contains at least µ 1 n 3m copies of K 3 (m) of type (m + s, m + t).
Claim 4. Given integers 0 s, t m with s + t = m, there exists µ 1 > 0 such that the following holds: If |E 0 | < γ 4 n 2 , then H contains at least µ 1 n 3m copies of K 3 (m) of the same type either (2m + s, t) or (t, 2m + s).
We claim that either d E 1 (y) 3γ 2 n or d E 1 (y) |X| − 3γn for all y ∈ Y . Fix y ∈ Y . Let e y be the number of edges x 1 x 2 y of the form XXY such that exactly one of {x 1 y, x 2 y} belongs to E 1 . On one hand, we have since for each x ∈ N E 1 (y), there are at least (1/2 − γ − γ 2 )n − d E 1 (y) edges xx y of the form XXY with x ∈ N E 0 (y) ∪ N E 2 (y) and |X| (1/2 + γ + γ 2 )n. On the other hand, we have e y |X| · d E 0 (y) + γ 2 n · d E 2 (y) 2γ 2 n|X|, since d X (x y) < γn 2 for x y ∈ E 2 , and the last inequality holds since d E 0 (y) γ 2 n and d E 2 (y) |X|. Therefore, we have Solve the inequality we have either d E 1 (y) 3γ 2 n or d E 1 (y) |X| − 3γn for all y ∈ Y .
On the other hand, if the number of pairs Next, we claim that there are at least ( such that the electronic journal of combinatorics 27(3) (2020), #P3.47 d X (x 1 x 2 ) γn. In fact, the third inequality holds since d X (xy) |X| for any xy ∈ E 0 ∪ E 1 , d X (xy) < γ 2 n for xy ∈ E 2 and |E 0 | < γ 4 n 2 , |E 1 | n 2 8 and |E 2 | n 2 4 ; the last inequality holds since (1/2 − 3γ)n |X| (1/2 + 3γ)n. Since As what we have done in the proof of Claim 3, define an auxiliary 4-graph G 2 as follows. Let V (G 2 ) = X ∪ X ∪ Y , where X is a copy of X; for x ∈ X , x 1 , x 2 ∈ X, y ∈ Y , x x 1 x 2 y ∈ E(G 2 ) if and only if x x 1 x 2 , x 1 x 2 y ∈ H. Hence, n < |V (G 2 )| = n + |X| < 2n, and By Lemma 3.5, there exists a positive constant µ 2 such that G 2 contains at least µ 2 n 4m copies of K 4 (m). As the same argument shown in the proof of Claim 3, H contains at least µ 2 n 3m copies of K 3 (m) of type (2m + s, t) for some positive µ 2 .
This completes the proof of Lemma 2.1.

Extremal case
In this section, we prove Lemmas 2.3 and 2.4. Let G and H be two k-graphs on the same vertex set V and let G \ H be the graph (V, E(G) \ E(H)). Suppose that |V | = n and 0 α 1, we say a vertex v ∈ H α-good with respect to G if d G\H (v) αn k−1 , otherwise call it α-bad. We call H α-good with respect to G if all of vertices in H are α-good with respect to G. First we deal with a special case when H is α-good with respect to the extremal graph. We need a lemma from [13] which follow with some extra work from a perfect packing theorem of Lu and Székely [22]. Given V = A ∪ B, let D[A, B] be the k-graph on V consisting of all edges of type AB k−1 .
Lemma 4.1 (Lemma 6.1 in [13]). Let K be a complete k-partite k-graph of order t with the first part of size a 1 . Given 0 < ρ 1/m and a sufficiently large integer n, suppose H is a k-graph on n ∈ tZ vertices with a partition of V (H) = X ∪Y such that a 1 |Y | = (t−a 1 )|X|. Furthermore, assume that H is ρ-good with respect to D[X, Y ]. Then H contains a Kfactor. Remark: Note that, in the above proof, the K 3 (m)-factors M 1 and M 2 have the following property: (1) Each member in M 1 (resp. M 2 ) has type (m, 2m) (resp. (3m, 0)) with respect to the partition A ∪ B, and (2) both |M 1 |(∼ n 4 ) and |M 2 |(∼ n 12 ) are large enough. The following classical result [16] also will be used.

Proofs of Lemmas 2.3 and 2.4
Since H is γ-extremal, there is a partition V = A ∪ B such that |A| |B| n/2 and . Now move all A-acceptable vertices into A and B-acceptable vertices into B, we get a new partition V = A ∪ B with the property that 1) n/2 − γ 1 n |A |, |B | n/2 + γ 1 n (since |A 0 ∪ B 0 | γ 1 n); 2) H γ 2 -contains B[A , B ] for some constant γ 2 γ 1 . Moreover, we can partition A into A 1 , A 2 so that: 50 . Our strategy is to find vertex-disjoint K 3 (m)-tiling K 1 , K 2 , K 3 , K 4 in H so that the union of them is a K 3 (m)-factor of H, in which K 1 is so-called 'parity breaking' copies dealing with the case |B | ≡ 0 (mod 2m), K 2 covers all vertices in A 2 ∪ B 2 , and K 3 is used to guarantee the divisibility condition required by Lemma 4.2 after removing the vertices covered by K 1 and K 2 . Furthermore, K 1 , K 2 , K 3 are all small enough such that the graph obtained by deleting K 1 , K 2 , K 3 is γ 3 -good for some constant γ 3 . Finally, we apply Lemma 4.2 to obtain K 4 .
In Claims 5 and 6, we show that such 'parity breaking' copies of K 3 (m) (resp. K 3 m,m ) do exist.
Proof of Claim 5: If we can find a copy of K 3 (m) of type (m + 1, 2m − 1) or (3m − 1, 1) avoiding any given vertex set W ⊂ V with |W | C for some constant C 6m 2 , then we can greedily find 2m − 1 disjoint copies of K 3 (m) of desired type because we always can find a new copy of K 3 (m) avoiding the vertices of copies of K 3 (m) we have found (since C 6m 2 ). So the rest of the proof is to show the statement is true. Choose any vertex set W ⊂ V with |W | C for some constant C 6m 2 . We split the proof into two cases according to the size of B .
First assume that |B | n/2. For any a ∈ A , b ∈ B , we have |N H (ab) ∩ A | m 1/m n 1−1/m since δ 2 (H) n/2 + m 1/m n 1−1/m . Construct an auxiliary bipartite graph G as follows: set V (G) = A ∪ B and E(G) consists of all pairs ab with a ∈ A , b ∈ B and  A , B )) contains a matching of order m − 1, choose such a matching a 2 b 2 , . . . , a m b m . So the subgraph induced by {a 1 , a 1 , a 2 , . . . , a m } ∪ {b 1 , . . . , b m } ∪ {b 2 , . . . , b m } of H contains a copy of K 3 m,m of type (m + 1, 2m − 1). Now assume |N H (a 1 a 2 ) ∩ B | < 2γ 1 n for any a 1 a 2 ∈ A 2 . Then |N H (a 1 a 2 ) ∩ A | > n/2 − 2 − 2γ 1 n. Let F be the spanning subgraph consisting of all the edges of type A A B of H. We claim that if there is some b ∈ B such that |F b | > 2mγ 1 n, then H contains a copy of K 3 m,m of type (3m − 1, 1). In fact, assume that there is some b ∈ B with |F b | > 2mγ 1 n. First, suppose that F b contains a matching of size m. Let a 1 a 1 , . . . , a m a m be a matching of F b . Since contains a matching of order m − 1, choose such a matching a 2 a 2 , . . . , a m a m . Therefore, the subgraph of H induced by {a 1 , . . . , a m } ∪ {a 2 , a 2 , . . . , a m , a m } ∪ {a, b} contains a copy of K 3 m,m of type (3m − 1, 1), as desired. So the rest of the case is to show that such a vertex b ∈ B with |F b | 2mγ 1 n does exist.