Construction of S ( 3 ) ( 2, 3 ) -Designs of Any Index

: Let H ( 3 ) be a uniform hypergraph of rank 3. A hyperstar S ( 3 ) ( 2,3 ) of centre C = { x , y } is a 3-uniform hypergraph with three hyperedges, all having the centre C = { x , y } in common, with x and y of degree 3 and the remaining vertices of degree 1. In this paper, we determine the spectrum of S ( 3 ) ( 2,3 ) - designs for any index λ .


Introduction
Let K (h) v = (X, E ) be the complete hypergraph, uniform of rank h, defined on the vertex set X = {x 1 , x 2 , . . ., x v }.This means that E is the collection of all the subsets of X whose cardinality is h; we will call a set of cardinality h an h-subset.
Let H (h) be a subhypergraph of K (h) v .An H (h) -design, or also a design of type H (h) or a system of H (h) , with order v and index λ, is a pair Σ = (X, B), where X is a finite set of cardinality v, whose elements are called vertices, and B is a collection of hypergraphs over X, called blocks, all isomorphic to H (h) , under the condition that every h-subset of X is a hyperedge of exactly λ hypergraphs of the collection B. An H (h) -design, of order v and index λ, is also called an H (h) -decomposition of λK (h) v (see, for example, [1][2][3][4][5]).It is important to note that in the definition of index λ, we do not require that the blocks be distinct.That is, in a given example, a block may be repeated as many as λ times.
In what follows, we will indicate by Spectrum(H (h) ) the spectrum of the corresponding H (h) -designs, i.e., the set of all positive integers v such that there exist H (h) -designs of order v with blocks isomorphic to H (h) .Furthermore, we shall call:

•
Hyperstar S (h) (r, s) the h-uniform hypergraph with s hyperedges and order (h − r)s + r, such that all the edges have in common exactly the same r vertices, which form its centre, and all the vertices of the centre have degree s; • Hyperpath P (h) (r, v) the h-uniform hypergraph, with v vertices and m hyperedges E 1 , E 2 , . . . ,E m which can be ordered in such a way that Following this notation, the symbols P (h) (h − 1, h + 1) and S (h) (h − 1, 2) define the same class of isomorphism of hypergraphs.We refer to [6] for details.Some results on P (h) (r, v)-designs have been proven in refs.[7,8].
In the literature, we can find many results on graph decompositions for simple graphs.The results on hypergraph decompositions are limited to small hypergraphs of small uniformity or to a limited class of hypergraphs (see [9]).
In refs.[10,11], the authors introduced the notion of edge-balanced designs and completely determined the spectrum of edge-balanced S (3) (2, 3)-designs.Here, we drop the hypothesis of balanced edges.In refs.[1], the authors solved the problem of finding a used for other values of h and other hypergraphs H in the case that λ is always equal to 1 and H = H 1 = {{a, b, c}, {a, b, d}} and H = H 2 = {{a, b, c}, {a, b, d}, {a, c, d}}.The methods and tools used are standard, such as methods of differences and composition methods.
Here, we study another H-decomposition of λK The study of hypergraph designs has become an important research area of combinatorial design.In this field, the focus has always been on construction techniques.In general, in the literature, B is a multiset (see [12]).In this paper, we provide an exhaustive result about the spectrum of S (3) (2, 3)-designs for any index λ with the further condition that in the system, all the blocks are distinct; i.e., we find the cardinality of X, where (X, B) is a λS (3) (2, 3)-design and B is a set and not a multiset for λ = 1, 2, 3.For λ > 3, we deal with B as a multiset.We end of the paper by posing a question regarding the general cases for further investigation.

Preliminary Results
In the following, we will use hypergraph terminology and will indicate the n-star graph, i.e., the connected graph with n vertices and a centre vertex of degree n − 1, by S (2) (1, n − 1).
The following is well known, see [13].
Theorem 1.There exists an S (2) (1, k)-design of order v if and only if v(v − 1) ≡ 0 mod 2k and v ≥ 2k From this, the following result.

The Spectrum of S
In this section, we determine the spectrum of S (3) (2, 3)-designs with index λ = 1; that is, the designs considered will always be of index 1.Let X = {x, y, z 1 , z 2 , z 3 } be a set of vertices, {x, y} be the centre and E = {{x, y, z i } : i = 1, 2, 3} be the set of hyperedges.Such a hypergraph will be denoted by [(x, y) First of all, we prove the following results: ) is an S (3) (2, 3)-design of order v and index λ, then: 2.

1.
Since the blocks of Σ = (X, B) contain, among their hyperedges, all the triples of X with multiplicity λ, and each block contains three of them, it follows that: As necessary, the only factor in the numerator which is a multiple of 3 must be a multiple of 9.
We recall the following definitions.
) is an H-design and x ∈ X, we call degree of the vertex x the number d(x) of blocks of B containing x; for any x, y ∈ X, x ̸ = y, we call degree of edge (x, y) the number d(x, y) of blocks of B containing edge (x, y).
Definition 2. An H (3) -design is said to be balanced if the degree d(x) of each vertex x ∈ X is a constant.Definition 3.An H (3) -design is called edge-balanced if, for any x, y ∈ X, x ̸ = y, the degree d(x, y) is constant.

Definition 4 ([15]
).A Steiner quadruple system is a pair (X, B), where X is a finite set and B is a collection of four-subsets of X (called blocks) such that any three-subset of X belongs to exactly one block of B. The number |X| = v is called the order of the quadruple system and it is denoted by SQS(v).
Proof.Let us denote by X the set {0, We also observe that in every block, each pair is repeated one time, that is, Γ = (X \ {0}, C) is edge-balanced.From Γ, we construct the family B of hypergraph S (3) (2, 3) by appending the vertex 0 to the blocks C ∈ C in the following way: One can verify that Σ = (X, B) is an S (3) (2, 3)-design of order v = 9.
Denote by [(x), y, z, t] the element of D with vertices {x, y, z, t} and centre {x}, we construct the family D ′ of blocks isomorphic to S (3) (2, 3) by adding ⋆ to the centre of [(x), y, z, t], i.e., D ′ = {[(⋆, x), y, z, t]] : [(x), y, z, t] ∈ D} We show that the system Given any triple {x, y, z} ⊆ X ′ , one can verify that there exists at least one block of B ∪ D ′ containing it.Indeed, if {x, y, z} is a triple of X, then there exists a block of B containing it as a hyperedge.If {x, y, ⋆} is a triple of X ′ , then there exists a block of D containing the pair {x, y} ⊆ X and, therefore, there exists a block of D ′ containing the triple {x, y, ⋆}.
Since Σ 1 and Σ 2 are S (3) (2, 3)-designs with λ = 1, we have that: which is the exact number of blocks in any S (3) (2, 3)-design of order We summarize all the previous results in the main theorem of this section.
In what follows, we will denote by Z n the additive cyclic group of order n whose elements are listed from 1 to n, and by [(x), y 1 , . . ., y n ] the star graph with centre x and edges xy i , i.e., S (2) (1, n).We recall some known results that we will use in the following.

Proof.
It is immediately clear that there exists a (unique) 3S (2) (1, 3)-design of order 4 and there are not any for v ≤ 3. We now proceed by induction on the number of vertices v; given v, consider the table A with three rows and v columns whose elements are a ij = i + j − 1 for i ∈ {1, 2, 3}, j ∈ Z v = {1, . . ., v} and the sum i + j − 1 is performed modulo v.It is evident that the columns of A are all distinct and each element of Z v appears three times.Denote by a •j the j-th column of a and consider the graph (⋆)a •j , which is isomorphic to S (2) (1,3).Denote by F the set of these graphs.
6.The Spectrum of S (3) (2, 3)-Designs of Index λ > 3 For the sake of completeness, in this section, we deal with the problem of existence of λS (3) (2, 3)-designs for any index λ, where the blocks are repeated with multiplicity λ.
In Section 3, we have already proven that if v ≡ 0, 1, 2 mod 9, then there exists a S (3) (2, 3)-design (X, B) of order v and index λ = 1.If repetitions of blocks are allowed, to construct a λS (3) (2, 3)-design of order v, it is sufficient to consider the design (X, λB), where λB is the uniform multiset with underlying set B and multiplicity λ.
Similar arguments apply to the case λ ≡ 0 mod 3 and v ≥ 5: in Section 5, we proved that there exists a 3S (3) (2, 3)-design (X, B) of order v ≥ 5; in order to construct a λS (3) (2, 3)design, it is sufficient to consider the design (X, λ 3 B), where λ 3 B is the uniform multiset with underlying set B and multiplicity λ 3 .To conclude our paper, we suggest the following open problem: Determining the spectrum of S (3) (2, n) for any positive integer n.

Author Contributions:
Writing-original draft: A.C., M.G. and E.G.; writing-review and editing: A.C., M.G. and E.G.All authors have read and agreed to the published version of the manuscript.Funding: Causa, Gionfriddo and Guardo were partially supported by the Università degli Studi di Catania, "PIACERI 20/22 Linea di intervento 2".Guardo was supported by the project "0dimensional schemes, Tensor Theory and applications"-PRIN 2022-Finanziato dall'Unione europea-Next Generation EU-CUP: E53D23005670006.Data Availability Statement: Data are contained within the article.