On maximum packings of λ-fold complete 3-uniform hypergraphs with triple-hyperstars of size 4

A symmetric triple-hyperstar is a connected, 3-uniform hypergraph where, for some edge {a, b, c}, vertices a, b, and c all have degree k > 1 and all other edges contain exactly 2 vertices of degree 1. Let H denote the symmetric triple-hyperstar with 4 edges and, for positive integers λ and v, let K (3) v denote the λ-fold complete 3-uniform hypergraph on v vertices. We find maximum packings of λK v with copies of H .


Introduction
A hypergraph H consists of a finite, nonempty set V of vertices and a finite collection E = {e 1 , e 2 , . . . , e m } of nonempty subsets of V called hyperedges or simply edges. For a given hypergraph H, we use V (H) and E(H) to denote the vertex set and the edge set (or multiset) of H, respectively. We call |V (H)| and |E(H)| the order and size of H, respectively. A hypergraph H is simple if no edge appears more than once in E(H). If for each e ∈ E(H) we have |e| = t, then H is said to be t-uniform. Thus t-uniform hypergraphs are generalizations of the concept www.ejgta.org of a graph (where t = 2). Graphs with repeated edges are often called multigraphs. If H is a simple hypergraph and λ is a positive integer, then λ-fold H, denoted λ H, is the multi-hypergraph obtained from H by repeating each edge exactly λ times. The hypergraph with vertex set V and edge set the set of all t-element subsets of V is called the complete t-uniform hypergraph on V and is denoted by v is called the λ-fold complete t-uniform hypergraph of order v and is used to denote any hypergraph isomorphic to λ K (t) V . When t = 2, we will use λ K v in place of λ K (2) v . Similarly, if λ = 1, then we will use K v . If H is a subhypergraph of H, then H \ H denotes the hypergraph obtained from H by deleting the edges of H . We may refer to H \ H as the hypergraph H with a hole H . The vertices in H may be referred to as the vertices in the hole.
A commonly studied problem in combinatorics concerns decompositions of graphs or multigraphs into edge-disjoint subgraphs. A decomposition of a multigraph K is a set ∆ = {G 1 , G 2 , . . . , G s } of subgraphs of K such that {E(G 1 ), E(G 2 ), . . . , E(G s )} is a partition of E(K). If each element of ∆ is isomorphic to a fixed graph G, then ∆ is called a G-decomposition of K. If exactly one element L ∈ ∆ is not isomorphic to G, then ∆ is called a G-packing of K with leave L. Such a G-packing is maximum if no other possible G-packing of K has a leave of a smaller size than that of L. Clearly, if |E(L)| < |E(G)|, then the G-packing is maximum. Moreover, a G-decomposition of K can be viewed as a maximum G-packing with an empty leave.
A G-decomposition of λ K v is also known as a G-design of order v and index λ. A K k -design of order v and index λ is usually known as a 2-(v, k, λ) design or as a balanced incomplete block design of index λ or a (v, k, λ)-BIBD. The problem of determining all v for which there exists a G-design of order v is of special interest (see [1] for a survey).
The notion of decompositions of graphs naturally extends to hypergraphs. A decomposition of a hypergraph K is a set ∆ = {H 1 , H 2 , . . . , H s } of subhypergraphs of K such that {E(H 1 ), E(H 2 ), . . . , E(H s )} is a partition of E(K). Any element of ∆ isomorphic to a fixed hypergraph H is called an H-block. If all elements of ∆ are H-blocks, then ∆ is called an H-decomposition of K. If exactly one element L ∈ ∆ is not an H-block, then ∆ is called an H-packing of K with leave L, where we again define such a packing to be maximum if L has the fewest edges possible. An H-decomposition of λ K (t) v is called an H-design of order v and index λ. The problem of determining all v for which there exists an H-design of order v and index λ is called the λ-fold spectrum problem for H-designs.
A K (t) k -design of order v and index λ is a generalization of 2-(v, k, λ) designs and is known as a t-(v, k, λ) design or simply as a t-design. A summary of results on t-designs appears in [16]. A t-(v, k, 1) design is also known as a Steiner system and is denoted by S(t, v, k) (see [9] for a summary of results on Steiner systems). Keevash [15] has recently shown that for all t and k the obvious necessary conditions for the existence of an S(t, k, v)-design are sufficient for sufficiently large values of v. Similar results were obtained by Glock, Kühn, Lo, and Osthus [10,11] and extended to include the corresponding asymptotic results for H-designs of order v for all uniform hypergraphs H. These results for t-uniform hypergraphs mirror the celebrated results of Wilson [24] for graphs. Although these asymptotic results assure the existence of H-designs for sufficiently large values of v for any uniform hypergraph H, the spectrum problem has been settled for very few hypergraphs of uniformity larger than 2.
In the study of graph decompositions, a fair amount of the focus has been on G-decompositions of K v where G is a graph with a relatively small number of edges (see [1] and [6] for known results). Some authors have investigated the corresponding problem for 3-uniform hypergraphs. For example, in [4], the 1-fold spectrum problem is settled for all 3-uniform hypergraphs on 4 or fewer vertices. More recently, the 1-fold spectrum problem was settled in [5] for all 3-uniform hypergraphs with at most 6 vertices and at most 3 edges. In [5], they also settle the 1-fold spectrum problem for the 3-uniform hypergraph of order 6 whose edges form the lines of the Pasch configuration. Authors have also considered H-designs where H is a 3-uniform hypergraph whose edge set is defined by the faces of a regular polyhedron. Let T , O, and I denote the tetrahedron, the octahedron, and the icosahedron hypergraphs, respectively. The hypergraph T is the same as K 4 , and its spectrum was settled in 1960 by Hanani [12]. In another paper [13], Hanani settled the spectrum problem for O-designs and gave necessary conditions for the existence of I-designs. The 1-fold spectrum problem is also settled for a type of 3-uniform hyperstars which is part of a larger class of hypergraphs known as delta-systems. For a positive integer m, let S v are given in [22] for m ∈ {4, 5, 6} and settled in [19] for any m. Some results on maximum S v are given in [20]. Perhaps the best known general result on decompositions of complete t-uniform hypergraphs is Baranyai's result [3] on the existence of 1-factorizations of K (t) mt for all positive integers m. There are, however, several articles on decompositions of complete t-uniform hypergraphs (see [2] and [21]) and of t-uniform t-partite hypergraphs (see [17] and [23]) into variations on the concept of a Hamilton cycle. There are also several results on decompositions of 3-uniform hypergraphs into structures known as Berge cycles with a given number of edges (see for example [14] and [18]). We note however that the Berge cycles in these decompositions are not required to be isomorphic.
In this paper we are interested in maximum H-packings of λ K v , where H is a 3-uniform symmetric triple-hyperstar with 4 edges. A triple-hyperstar is a connected 3-uniform hypergraph where, for some edge {a, b, c}, vertices a, b, and c all have degree greater than 1 and all other edges contain exactly two vertices of degree 1. That is, if the degrees of vertices a, b, and c in the triple-hyperstar are m 1 + 1, m 2 + 1, and m 3 + 1, respectively, then the removal of edge {a, b, c} would result in the hypergraph consisting of three components, namely S   Let Figure 1. Here we show that for all v ≥ 9 and λ ≥ 1, there exists a maximum H-packing of λ K (3) v where the leave has fewer than 4 edges.

Additional Notation and Terminology
Let Z n denote the group of integers modulo n. We next define some notation for certain types of 3-uniform hypergraphs. www.ejgta.org Let U 1 , U 2 , U 3 be pairwise disjoint sets. The hypergraph with vertex set U 1 ∪ U 2 ∪ U 3 and edge set consisting of all 3-element sets having exactly one vertex in each of U 1 , U 2 , U 3 is denoted by K The hypergraph with vertex set U 1 ∪ U 2 and edge set consisting of all 3-element sets having at most 2 vertices in each of U 1 , U 2 is denoted by L

Decompositions and Packings of Simple Hypergraphs
We begin by giving necessary conditions for the existence of an H-decomposition of K (3) v . An obvious necessary condition is that 4 must divide the number of edges in K (3) v , and thus we must have v ≡ 0, 1, 2, 4, or 6 (mod 8). Since K We intend to prove that the above conditions are sufficient by showing how to construct H- Our constructions are dependent on the many small examples given in the Appendix. We begin by proving a lemma that is fundamental to our constructions. Lemma 2. Let n, x, and r be nonnegative integers such that nx + r ≥ 3. There exists a decomposition of K (3) nx+r that is comprised of isomorphic copies of each of the following under the given conditions: Furthermore, if x ≥ 1 and r ≥ 3, then the decomposition contains exactly one isomorphic copy of K Proof. If x ∈ {0, 1}, the decomposition is trivial. Similarly, if n = 0, then r ≥ 3, and the result is trivial because K n,n , and K (3) n,n,n are all empty (i.e., contain no edges). For the remainder of the proof, we assume that x ≥ 2 and n ≥ 1.
nx+r results from the fact that the complete 3-uniform hypergraph on the vertex set V 0 ∪ V 1 ∪ · · · ∪ V x , which is nx + r vertices, can be viewed as the (edge-disjoint) union In addition, if r ≥ 3, the single isomorphic copy of K We now give our main results.    where the leave has fewer than four edges.
Proof. If v ≡ 0, 1, 2, 4, or 6 (mod 8), then the result follows from the H-decomposition result in Theorem 3, which translates to a maximum H-packing with an empty leave. Hence, we need only consider when v ≡ 3, 5, or 7 (mod 8). Let v = 8x + r where x ≥ 1 and r ∈ {3, 5, 7}. By Lemma 2 it suffices to find • a maximum H-packing of K We note that an H-decomposition of K 11 with a leave consisting of the single edge in the hole, which is necessarily then a maximum H-packing of K
Next, we settle the decomposition and maximum packing results for some small values of λ. Theorem 6. Let v ≥ 9 be an integer. There exists an H-decomposition of 2-fold K Proof. If v ≡ 0, 1, 2, 4, or 6 (mod 8), then the result follows from 2 copies of an H-decomposition of K 13 with a leave consisting of two edges that share a single vertex and a maximum H-packing, say ∆ 2 , of K 13 with a leave consisting of two vetexdisjoint edges. Let L 1 and L 2 be the leaves of ∆ 1 and ∆ 2 , respectively. Without loss of generality, we may assume that Now, let L be the hypergraph with edge set E(L 1 ) ∪ E(L 2 ). Hence, L is isomorphic to H, and the (multi-)set is a collection of H-blocks such that each edge of K 13 is represented exactly twice. Therefore, we have an H-decomposition of 2 K 13 . Now, let v = 8x + 5 where x ≥ 2. By Lemma 2 it suffices to find H-decompositions of (2-fold) K  Proof. If v ≡ 0, 1, or 2 (mod 4), then the result follows from the H-decomposition result in Theorem 6, which translates to a maximum H-packing with an empty leave. Hence, we need only consider when v ≡ 3 (mod 4).
First, we consider when v = 11. Let ∆ 1 and ∆ 2 be maximum H-packings of K 11 with leaves L 1 and L 2 , respectively, which exist by Example 17. Now, let L be the hypergraph with edge (multi-)set E(L 1 ) ∪ E(L 2 ). Hence, L consists of two edges. In fact, we further note that L can be any hypergraph with two edges, including 2 K 3 . Hence, the (multi-)set 15 where the leaves consist of three disjoint edges. Let ∆ 1 and ∆ 2 be such H-packings of K 15 with leaves L 1 and L 2 , respectively. Without loss of generality, we may assume that Now, let L be the hypergraph with edge set E(L 1 ) ∪ E(L 2 ). We note that L is decomposable into copies of K and H. That is, if we let L be the hypergraph with edge set {v 4 , v 5 , v 6 }, {v 7 , v 8 , v 9 } , then L \ L is isomorphic to H, and the (multi-)set 15 with a leave, L , consisting of two (disjoint) edges. Now, let v = 8x + r where x ≥ 2 and r ∈ {3, 7}. By Lemma 2 it suffices to find • a maximum H-packing of (2-fold) K We already have the maximum H-packing results. Also, we note that K Proof. If v ≡ 0, 1, 2, 4, or 6 (mod 8), then the result follows from the H-decomposition result in Theorem 3, which translates to a maximum H-packing with an empty leave. Hence, we need only consider when v ≡ 3, 5, or 7 (mod 8).
First, we consider when v = 11. Let ∆ 1 be a maximum H-packing of K 11 with leave L 1 consisting of a single edge, which exists by Example 17, and let ∆ 2 be a maximum H-packing of 2 K 11 with leave L 2 consisting of two edges, which exists by Theorem 7, Now, let L be the hypergraph with edge (multi-)set E(L 1 ) ∪ E(L 2 ). Hence, L consists of three edges. In fact, we further note that L can be any hypergraph with three edges, including 3 K 3 . Hence, the (multi-)set 11 with a leave, L , consisting of three edges. Second, we consider when v = 13. Let ∆ 1 be a maximum H-packing of K 15 with leave L 1 consisting of a three vertex-disjoint edges, which exists by Example 19, and let ∆ 2 be a maximum H-packing of 2 K 15 with leave L 2 consisting of two vertexdisjoint edges, which exists by Theorem 7, Without loss of generality, we may assume that Now, let L be the hypergraph with edge set E(L 1 ) ∪ E(L 2 ). We note that L is decomposable into copies of K 3 and H. That is, if we let L be the hypergraph with the single edge {v 4 , v 5 , v 6 }, then L \ L is isomorphic to H, and the (multi-)set 15 with a leave, L , consisting of one edges. Now, let v = 8x + r where x ≥ 2 and r ∈ {3, 5, 7}. By Lemma 2 it suffices to find • a maximum H-packing of (3-fold) K We already have the maximum H-packing results. Also, we note that K Proof. If v ≡ 0, 1, or 2 (mod 4), then the result follows from 2 copies of an H-decomposition v , which exists by Theorem 6. Hence, we need only consider when v ≡ 3 (mod 4).
First, we consider when v = 11. For i ∈ {1, 2, 3, 4}, let ∆ i be a maximum H-packing of K with leave L i consisting of a single edge, which exists by Example 17, Without loss of generality, we may assume that Now, let L be the hypergraph with edge set E(L 1 ) ∪ E(L 2 ) ∪ E(L 3 ) ∪ E(L 4 ). Hence, L is isomorphic to H, and the (multi-)set is a collection of H-blocks such that each edge of K 11 is represented exactly four times. Therefore, we have an H-decomposition of 4 K 15 with leave L 2 consisting of a single edge, which exists by Theorem 8, Without loss of generality, we may assume that Now, let L be the hypergraph with edge set E(L 1 ) ∪ E(L 2 ). Hence, L is isomorphic to H, and the (multi-)set is a collection of H-blocks such that each edge of K 15 is represented exactly four times. Therefore, we have an H-decomposition of 4 K 15 . Now, let v = 8x + r where x ≥ 2 and r ∈ {3, 7}. By Lemma 2 it suffices to find Hdecompositions of (4-fold) K  Finally, we show that the necessary conditions for the existence of an H-decomposition of λ-fold K • if gcd(λ, 4) = 1, then v ≡ 0, 1, 2, 4, or 6 (mod 8); • if gcd(λ, 4) = 2, then v ≡ 0, 1, or 2 (mod 4); • if gcd(λ, 4) = 4, then v ≥ 9.
Proof. The necessary conditions are established in Lemma 5. For sufficiency, we consider the following cases.
Case 1. λ ≡ 0 (mod 4) Let λ = 4t for some positive integer t. Then the result follows from t copies of an H-decomposition of 4 K v , which exists by Theorem 3. Case 3. λ ≡ 2 (mod 4) Since gcd(λ, 4) = 2, we have that v ≡ 0, 1, or 2 (mod 4). Let λ = 4t + 2 for some nonnegative integer t. Then the result follows from t copies of an H-decomposition of 4 K Theorem 11. If v ≥ 9 is an integer, then there exists a maximum H-packing of λ-fold K (3) v where the leave has fewer than four edges.
Then an H-decomposition of K consists of the orbits of the H-blocks in B 1 under the action of the map j → j + 1 (mod 12) along with the H-blocks in B 2 .