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

Abstract

Let $H^{\left(3\right)}$ be a uniform hypergraph of rank 3. A hyperstar $S^{\left(3\right)}(2,3)$ 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 $S^{\left(3\right)}(2,3)$-designs for any index $\lambda$.

1. Introduction

Let $K_v^{\left(h\right)}=(X,\mathcal{E})$ be the complete hypergraph, uniform of rank h, defined on the vertex set $X=\{x_1,x_2,\dots ,x_v\}$. This means that $\mathcal{E}$ 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 $H^{\left(h\right)}$ be a subhypergraph of $K_v^{\left(h\right)}$. An $H^{\left(h\right)}$-design, or also a design of type $H^{\left(h\right)}$ or a system of $H^{\left(h\right)}$, with order v and index $\lambda$, is a pair $\Sigma =(X,\mathcal{B})$, where X is a finite set of cardinality v, whose elements are called vertices, and $\mathcal{B}$ is a collection of hypergraphs over X, called blocks, all isomorphic to $H^{\left(h\right)}$, under the condition that every h-subset of X is a hyperedge of exactly $\lambda$ hypergraphs of the collection $\mathcal{B}$. An $H^{\left(h\right)}$-design, of order v and index $\lambda$, is also called an $H^{\left(h\right)}$-decomposition of $\lambda K_v^{\left(h\right)}$ (see, for example, [1,2,3,4,5]).

It is important to note that in the definition of index $\lambda$, we do not require that the blocks be distinct. That is, in a given example, a block may be repeated as many as $\lambda$ times.

In what follows, we will indicate by $\mathsf{Spectrum}\left(H^{\left(h\right)}\right)$ the spectrum of the corresponding $H^{\left(h\right)}$-designs, i.e., the set of all positive integers v such that there exist $H^{\left(h\right)}$-designs of order v with blocks isomorphic to $H^{\left(h\right)}$. Furthermore, we shall call:

• Hyperstar $S^{\left(h\right)}(r,s)$ 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 $P^{\left(h\right)}\left(r,v\right)$ the h-uniform hypergraph, with v vertices and m hyperedges $E_1,E_2,\dots ,E_m$ which can be ordered in such a way that $\left|E_i\cap E_{i+1}\right|=r$ and $E_i\cap E_j=\varnothing$ if $j\ge i+2$.

Following this notation, the symbols $P^{\left(h\right)}\left(h-1,h+1\right)$ and $S^{\left(h\right)}\left(h-1,2\right)$ define the same class of isomorphism of hypergraphs. We refer to [6] for details. Some results on $P^{\left(h\right)}\left(r,v\right)$-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 $S^{\left(3\right)}(2,3)$-designs. Here, we drop the hypothesis of balanced edges. In refs. [1], the authors solved the problem of finding a H-decomposition of $\lambda K_v^{\left(h\right)}$ for $h=3$ and $v=4$ and gave general methods which could be used for other values of h and other hypergraphs H in the case that $\lambda$ 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 $\lambda K_5^{\left(3\right)}$, where $H=\left\{\right\{a,b,c\},\{a,b,d\},\{a,b,e\left\}\right\}$ and $\lambda =1,2,3$.

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, $\mathcal{B}$ is a multiset (see [12]). In this paper, we provide an exhaustive result about the spectrum of $S^{\left(3\right)}\left(2,3\right)$-designs for any index $\lambda$ with the further condition that in the system, all the blocks are distinct; i.e., we find the cardinality of X, where $(X,\mathcal{B})$ is a $\lambda$$S^{\left(3\right)}\left(2,3\right)$-design and $\mathcal{B}$ is a set and not a multiset for $\lambda =1,2,3$. For $\lambda >3$, we deal with $\mathcal{B}$ as a multiset. We end of the paper by posing a question regarding the general cases for further investigation.

2. 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 $S^{\left(2\right)}(1,n-1)$.

The following is well known, see [13].

Theorem 1. 

There exists an $S^{\left(2\right)}(1,k)$-design of order v if and only if

$$v(v-1)\equiv 0mod2k\mathit{and}v\ge 2k$$

From this, the following result.

Corollary 1. 

There exists an $S^{\left(2\right)}(1,2)$-design of order v if and only if

$$v\equiv 0mod4\mathit{or}v\equiv 1mod4\mathit{and}v\ge 4$$

Corollary 2. 

There exists an $S^{\left(2\right)}(1,3)$-design of order v if and only if

$$v\equiv 0mod3\mathit{or}v\equiv 1mod3\mathit{and}v\ge 6$$

Regarding systems of $P^{\left(3\right)}\left(r,s\right)$, i.e., $P^{\left(3\right)}\left(r,s\right)$-designs, we have the following result.

Theorem 2. 

There exist $P^{\left(3\right)}\left(1,5\right)$-designs of order v if and only if

$$v\equiv 0mod2\mathit{and}v\ge 4\mathit{or}v\equiv 1mod4\mathit{and}v\ge 5$$

In what follows, we will denote the rank 3 uniform hypergraph $H=(X,\mathcal{E})$, where, e.g., $X=\left\{a,b,c,d\right\}$ and $\mathcal{E}=\left\{\left\{a,b,c\right\},\left\{a,b,d\right\}\right\}$, by $\left[\right(a,b),c,d]$.

Theorem 3. 

There exist $P^{\left(3\right)}(2,4)$-designs of order v if and only if $v\equiv 0$, or 1, or $2mod4$ and $v\ge 4$.

The previous results can be found in refs. [8,14].

3. The Spectrum of ${\mathit{S}}^{\left(\mathbf{3}\right)}\left(\mathbf{2},\mathbf{3}\right)$-Designs of Index 1

In this section, we determine the spectrum of $S^{\left(3\right)}\left(2,3\right)$-designs with index $\lambda =1$; that is, the designs considered will always be of index 1. Let $X=\left\{x,y,z_1,z_2,z_3\right\}$ be a set of vertices, $\{x,y\}$ be the centre and $\mathcal{E}=\left\{\{x,y,z_i\}:i=1,2,3\right\}$ be the set of hyperedges. Such a hypergraph will be denoted by $[(x,y),z_1,z_2,z_3]$.

First of all, we prove the following results:

Theorem 4. 

If $\Sigma =(X,\mathcal{B})$ is an $S^{\left(3\right)}(2,3)$-design of order v and index λ, then:

1. 

$\left|\mathcal{B}\right|=\lambda {\displaystyle \frac{v(v-1)(v-2)}{18}}$.

2. 

If $\lambda =1$, then $v\equiv 0$, or $v\equiv 1$, or $v\equiv 2mod9$, and $v\ge 9$.

Proof. 

Let $\Sigma =(X,\mathcal{B})$ be an $S^{\left(3\right)}(2,3)$-design of order v and index $\lambda$.

• Since the blocks of $\Sigma =(X,\mathcal{B})$ contain, among their hyperedges, all the triples of X with multiplicity $\lambda$, and each block contains three of them, it follows that:

  $$\left|\mathcal{B}\right|=\lambda ·\left(\genfrac{}{}{0pt}{}{v}{3}\right)·\frac{1}{3}=\lambda ·{\displaystyle \frac{v(v-1)(v-2)}{3·2·1}}·{\displaystyle \frac{1}{3}}=\lambda ·{\displaystyle \frac{v(v-1)(v-2)}{18}}$$
• 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.

Definition 1. 

If $\Sigma =(X,\mathcal{B})$ is an H-design and $x\in X$, we call degree of the vertex x the number $d\left(x\right)$ of blocks of $\mathcal{B}$ containing x; for any $x,y\in X$, $x\ne y$, we call degree of edge $(x,y)$ the number $d(x,y)$ of blocks of B containing edge $(x,y)$.

Definition 2. 

An $H^{\left(3\right)}$-design is said to be balanced if the degree $d\left(x\right)$ of each vertex $x\in X$ is a constant.

Definition 3. 

An $H^{\left(3\right)}$-design is called edge-balanced if, for any $x,y\in X$, $x\ne y$, the degree $d(x,y)$ is constant.

See [6,10,14] for more details.

Definition 4 

([15]). A Steiner quadruple system is a pair $(X,\mathcal{B})$, where X is a finite set and $\mathcal{B}$ is a collection of four-subsets of X (called blocks) such that any three-subset of X belongs to exactly one block of $\mathcal{B}$. The number $\left|X\right|=v$ is called the order of the quadruple system and it is denoted by $\mathrm{SQS}\left(v\right)$.

Theorem 5. 

If $\Sigma =(X,\mathcal{B})$ is an $\mathrm{SQS}\left(v\right)$ and

$$\mathcal{C}=\left\{\right[(y,z),x,t],[(x,t),y,z]:\{x,y,z,t\}\in \mathcal{B}\},$$

then $\Gamma =(X,\mathcal{C})$ is a $P^{\left(3\right)}\left(2,4\right)$-design of order v.

Proof. 

See ref. [8]. □

Theorem 6. 

There exist $S^{\left(3\right)}(2,3)$-designs of order $v=9$.

Proof. 

Let us denote by X the set $\{0,1,2,\dots ,8\}$. Using Theorem 5, one can verify that $\Gamma =(X\setminus \{0\},\mathcal{C})$ is a $P^{\left(3\right)}\left(2,4\right)$-design of order 8, where $\mathcal{C}$ is the following set

$$\begin{array}{cccccccc}\hfill \{& \left[\right(1,2),3,4],\hfill & & \left[\right(1,3),5,7],\hfill & & \left[\right(1,4),5,8],\hfill & & \left[\right(1,5),2,6],\hfill \\ & \left[\right(1,6),3,8],\hfill & & \left[\right(1,7),4,6],\hfill & & \left[\right(1,8),2,7],\hfill & & \left[\right(2,3),6,7],\hfill \\ & \left[\right(2,4),6,8],\hfill & & \left[\right(2,5),4,7],\hfill & & \left[\right(2,6),1,5],\hfill & & \left[\right(2,7),1,8],\hfill \\ & \left[\right(2,8),3,5],\hfill & & \left[\right(3,4),1,2],\hfill & & \left[\right(3,5),2,8],\hfill & & \left[\right(3,6),4,5],\hfill \\ & \left[\right(3,7),4,8],\hfill & & \left[\right(3,8),1,6],\hfill & & \left[\right(4,5),3,6],\hfill & & \left[\right(4,6),1,7],\hfill \\ & \left[\right(4,7),2,5],\hfill & & \left[\right(4,8),3,7],\hfill & & \left[\right(5,6),7,8],\hfill & & \left[\right(5,7),1,3],\hfill \\ & \left[\right(5,8),1,4],\hfill & & \left[\right(6,7),2,3],\hfill & & \left[\right(6,8),2,4],\hfill & & \left[\right(7,8),5,6]\}.\hfill \end{array}$$

We also observe that in every block, each pair is repeated one time, that is, $\Gamma =(X\setminus \{0\},\mathcal{C})$ is edge-balanced. From $\Gamma$, we construct the family $\mathcal{B}$ of hypergraph $S^{\left(3\right)}(2,3)$ by appending the vertex 0 to the blocks $C\in \mathcal{C}$ in the following way:

$$C=\left[\right(x,y),z,t]\mapsto B=\left[\right(x,y),z,t,0]$$

One can verify that $\Sigma =(X,\mathcal{B})$ is an $S^{\left(3\right)}(2,3)$-design of order $v=9$. □

In the following lemmas, we prove that given an $S^{\left(3\right)}(2,3)$-design of order $v=9h$, there exists an $S^{\left(3\right)}(2,3)$-design of order $v^{\prime }$, where $v^{\prime }\in \{v+1,v+2,v+9\}$.

Lemma 1. 

If $v=9h\in \mathsf{Spectrum}\left(S^{\left(3\right)}(2,3)\right)$, then $v^{\prime }=9h+1$ belongs to $\mathsf{Spectrum}\left(S^{\left(3\right)}(2,3)\right)$.

Proof. 

Construction from $v=9h$ to $v=9h+1$.

Let $\Sigma =(X,\mathcal{B})$ be an $S^{\left(3\right)}(2,3)$-design of order $v=9h$, $h\ge 1$. Let $★\notin X$, $X^{\prime }=X\cup \{★\}$.

Since $v=9h\in \mathsf{Spectrum}\left(S^{\left(2\right)}(1,2)\right)$ by Corollary 1, let $\Delta =(X,\mathcal{D})$ be an $S^{\left(2\right)}(1,2)$-design of order $v=9h$ defined over X.

Denote by $\left[\right(x),y,z,t]$ the element of $\mathcal{D}$ with vertices $\{x,y,z,t\}$ and centre $\left\{x\right\}$, we construct the family ${\mathcal{D}}^{\prime }$ of blocks isomorphic to $S^{\left(3\right)}(2,3)$ by adding ★ to the centre of $\left[\right(x),y,z,t]$, i.e.,

$${\mathcal{D}}^{\prime }=\left\{\left[\right(★,x),y,z,t]]:[\left(x\right),y,z,t]\in \mathcal{D}\right\}$$

We show that the system ${\Sigma }^{\prime }=(X^{\prime },\mathcal{B}\cup {\mathcal{D}}^{\prime })$, where $X^{\prime }=X\cup \{★\}$, is an $S^{\left(3\right)}(2,3)$-design of order $v^{\prime }=9h+1$.

Given any triple $\left\{x,y,z\right\}\subseteq X^{\prime }$, one can verify that there exists at least one block of $\mathcal{B}\cup {\mathcal{D}}^{\prime }$ containing it. Indeed, if $\{x,y,z\}$ is a triple of X, then there exists a block of $\mathcal{B}$ containing it as a hyperedge. If $\{x,y,★\}$ is a triple of $X^{\prime }$, then there exists a block of $\mathcal{D}$ containing the pair $\{x,y\}\subseteq X$ and, therefore, there exists a block of ${\mathcal{D}}^{\prime }$ containing the triple $\{x,y,★\}$.

Finally, since $\lambda =1$, we get

$$\left|\mathcal{B}\cup {\mathcal{D}}^{\prime }\right|=\left|\mathcal{B}\right|+\left|{\mathcal{D}}^{\prime }\right|={\displaystyle \frac{(9h+1)9h(9h-1)}{18}}$$

which is the exact number of blocks in any $S^{\left(3\right)}(2,3)$-design of order $v^{\prime }=9h+1$. □

Lemma 2. 

If $v=9h+1\in \mathsf{Spectrum}\left(S^{\left(3\right)}(2,3)\right)$, then $v^{\prime }=9h+2$ belongs to $\mathsf{Spectrum}\left(S^{\left(3\right)}(2,3)\right)$.

Proof. 

Construction from $v=9h+1$ to $9h+2$. Since $v=9h+1\in \mathsf{Spectrum}\left(S^{\left(2\right)}(1,3)\right)$ by Corollary 2, we can follow the same procedure as in Lemma 1. Finally, since $\lambda =1$, we have that:

$$\begin{array}{cc}\hfill |{\mathcal{B}}^{\prime }|& =\left|\mathcal{B}\right|+|{\mathcal{D}}^{\prime }|=\frac{(9h+1)\left(9h\right)(9h-1)}{18}+\frac{(9h+1)9h}{6}=\hfill \\ \hfill & =\frac{(9h+1)9h}{6}\left(\frac{9h-1}{3}+1\right)=\frac{(9h+2)(9h+1)9h}{18}\hfill \end{array}$$

 □

Lemma 3. 

If $v=9h\in \mathsf{Spectrum}\left(S^{\left(3\right)}(2,3)\right)$, then $v^{\prime }=9h+9$ belongs to $\mathsf{Spectrum}\left(S^{\left(3\right)}(2,3)\right)$.

Proof. 

Construction from $v=9h$ to $v=9h+9$.

Let ${\Sigma }_1=(X_1,{\mathcal{B}}_1)$ be an $S^{\left(3\right)}(2,3)$-design of order $v_1=9h$ and ${\Sigma }_2=(X_2,{\mathcal{B}}_2)$ be an $S^{\left(3\right)}(2,3)$-design of order $v_2=9$ such that $X_1\cap X_2=\varnothing$.

Since $v_1=9h\in \mathsf{Spectrum}\left(S^{\left(2\right)}(1,3)\right)$, let ${\Delta }_1=(X,{\mathcal{D}}_1)$ be an $S^{\left(2\right)}(1,3)$-design of order $v=9h$ defined on $X_1$, and define the following family of blocks isomorphic to $S^{\left(3\right)}(2,3)$:

$${\mathcal{C}}_2=\left\{[(a,x),y,z,t]:a\in X_2\mathrm{and}[\left(x\right),y,z,t]\in {\mathcal{D}}_1\right\}.$$

Similarly, since $v_2=9\in \mathsf{Spectrum}\left(S^{\left(2\right)}(1,3)\right)$, if ${\Delta }_2=(X,{\mathcal{D}}_2)$ is an $S^{\left(2\right)}(1,3)$-design of order $v_2=9$ defined on $X_2$, we can consider the following family of blocks isomorphic to $S^{\left(3\right)}(2,3)$:

$${\mathcal{C}}_1=\left\{[(x,a),b,c,d]:x\in X_1\mathrm{and}[\left(a\right),b,c,d]\in {\mathcal{D}}_2\right\}$$

The system $(X,{\mathcal{B}}^{\prime })=(X_1\cup X_2,{\mathcal{B}}_1\cup {\mathcal{B}}_2\cup {\mathcal{C}}_1\cup {\mathcal{C}}_2)$ is an $S^{\left(3\right)}(2,3)$-design of order $v=9h+9$.

Indeed, it is easy to see that for every triple contained in $X_1\cup X_2$, there exists at least a block of ${\mathcal{B}}^{\prime }$ containing it.

Since ${\Sigma }_1$ and ${\Sigma }_2$ are $S^{\left(3\right)}\left(2,3\right)$-designs with $\lambda =1$, we have that:

$$\left|{\mathcal{B}}^{\prime }\right|=\left|{\mathcal{B}}_1\right|+\left|{\mathcal{B}}_2\right|+\left|{\mathcal{C}}_1\right|+\left|{\mathcal{C}}_2\right|={\displaystyle \frac{(9h+9)(9h+8)(9h+7)}{18}}$$

which is the exact number of blocks in any $S^{\left(3\right)}(2,3)$-design of order $v=v_1+v_2=9h+9$. □

We summarize all the previous results in the main theorem of this section.

Theorem 7. 

There exist $S^{\left(3\right)}\left(2,3\right)$-designs of order v if and only if

$$v\equiv 0\mathit{or}v\equiv 1\mathit{or}v\equiv 2mod9\mathit{and}v\ge 9$$

4. The Spectrum of ${\mathit{S}}^{\left(\mathbf{3}\right)}(\mathbf{2},\mathbf{3})$-Designs of Index 2

In this section, we investigate the spectrum of $S^{\left(3\right)}\left(2,3\right)$-designs with index $\lambda =2$ (2$S^{\left(3\right)}\left(2,3\right)$-designs for short) with the further condition that in the system, all the blocks are distinct; i.e., we find the cardinality of X, where $(X,\mathcal{B})$ is a 2$S^{\left(3\right)}\left(2,3\right)$-design and $\mathcal{B}$ is a set and not a multiset. We prove that v belongs to $\mathsf{Spectrum}\left(2S^{\left(3\right)}\left(2,3\right)\right)$ iff $v\equiv 0,1,2mod9$ and $v\ge 5$.

In what follows, we will denote by ${\mathbb{Z}}_n$ the additive cyclic group of order n whose elements are listed from 1 to n, and by $[\left(x\right),y_1,\dots ,y_n]$ the star graph with centre x and edges $x{y}_i$, i.e., $S^{\left(2\right)}\left(1,n\right)$. We recall some known results that we will use in the following.

Lemma 4 

([15]). For every $v\equiv 2,4mod6,v>4$, there exists at least two disjoint Steiner quadruple systems (SQSs for short) ${\Sigma }_1=(X,{\mathcal{B}}_1)$ and ${\Sigma }_2=(X,{\mathcal{B}}_2)$ defined on the same set X.

Lemma 5. 

There exists a $P^{\left(3\right)}\left(2,4\right)$-design of order v and index 2 such that all the blocks are pairwise distinct.

Proof. 

Let ${\Sigma }_1=(X,{\mathcal{B}}_1)$ and ${\Sigma }_2=(X,{\mathcal{B}}_2)$ denote two disjoint $\mathrm{SQS}\left(v\right)$ as in Lemma 4, and ${\Gamma }_1=(X,{\mathcal{C}}_1)$ and ${\Gamma }_2=(X,{\mathcal{C}}_2)$ are two disjoint $P^{\left(3\right)}\left(2,4\right)$-designs obtained from ${\Sigma }_1$ and ${\Sigma }_2$, respectively, as in Theorem 5. One can verify that the system $\Gamma =(X,{\mathcal{C}}_1\cup {\mathcal{C}}_2)$ is a $P^{\left(3\right)}\left(2,4\right)$-design of order v and index 2 such that all the blocks are pairwise distinct. □

Thus, we get the following.

Theorem 8. 

For every v such that $v\equiv 2,4mod6$ and $v>4$, there exists at least one $P^{\left(3\right)}\left(2,4\right)$-design of order v and index 2 such that all the blocks are pairwise distinct.

In order to prove the result, we still need other preliminary lemmas.

Lemma 6. 

There exist 2$S^{\left(2\right)}\left(1,3\right)$-designs of order 4.

Proof. 

The number of blocks of 2$S^{\left(2\right)}\left(1,3\right)$-designs of order 4 must be 4; hence, there is a unique design whose blocks are the four stars which one can exhibit.  □

Lemma 7. 

There exist 2$S^{\left(2\right)}\left(1,3\right)$-designs of order 6.

Proof. 

The number of blocks of a 2$S^{\left(2\right)}\left(1,3\right)$-design of order 6 is 10. Let us consider the set of vertices X as $X={\mathbb{Z}}_5\cup \{★\}$ with ${\mathbb{Z}}_5$ equipped with the structure of a cyclic additive group. We can consider two blocks of the form $\left[\right(1),2,3,★]$ and $\left[\right(1),3,5,★]$ and their translations by $1\in {\mathbb{Z}}_5$, leaving ★ fixed.  □

Lemma 8. 

There exist 2$S^{\left(2\right)}\left(1,3\right)$-designs of order 9.

Proof. 

The number of blocks of a 2$S^{\left(2\right)}\left(1,3\right)$-design of order 9 is equal to 24, as seen in the following set:

$$\begin{array}{cccccccc}\hfill \{& \left[\right(0,4),1,8],\hfill & & \left[\right(1,5),0,2],\hfill & & \left[\right(2,6),3,1],\hfill & & \left[\right(3,7),4,2],\hfill \\ & \left[\right(4,7),5,3],\hfill & & \left[\right(5,8),6,4],\hfill & & \left[\right(6,1),7,5],\hfill & & \left[\right(7,1),8,6],\hfill \\ & \left[\right(8,4),3,7],\hfill & & \left[\right(0,3),2,7],\hfill & & \left[\right(1,4),3,8],\hfill & & \left[\right(2,5),5,0],\hfill \\ & \left[\right(3,6),5,1],\hfill & & \left[\right(4,7),6,8],\hfill & & \left[\right(5,8),7,3],\hfill & & \left[\right(6,0),8,4],\hfill \\ & \left[\right(7,1),0,5],\hfill & & \left[\right(8,2),1,6],\hfill & & \left[\right(0,3),4,5],\hfill & & \left[\right(1,4),5,6],\hfill \\ & \left[\right(2,5),6,7],\hfill & & \left[\right(3,6),7,8)],\hfill & & \left[\right(2,7),4,8],\hfill & & \left[\right(0,6),5,8]\}.\hfill \end{array}$$

 □

Lemma 9. 

Given a 2$S^{\left(2\right)}\left(1,3\right)$-design of order $v=3h$, there exists a 2$S^{\left(2\right)}\left(1,3\right)$-design of order $v^{\prime }=v+1=3h+1$.

Proof. 

Let $\Sigma =({\mathbb{Z}}_{3h},\mathcal{B})$ be a 2$S^{\left(2\right)}\left(1,3\right)$-design over ${\mathbb{Z}}_{3h}$ and denote by $\mathcal{C}$ the set of blocks of the form $\left[\right(★),j,j+1,j+2]$ and $\left[\right(★),j+1,j+2,j+3]$ with $j=1+3a\in {\mathbb{Z}}_{3h}$ for $0\le a\le h-1$. It is straightforward that $(\{★\}\cup {\mathbb{Z}}_{3h},\mathcal{B}\cup \mathcal{C})$ is a 2$S^{\left(2\right)}\left(1,3\right)$-design of order $3h+1$.  □

Lemma 10. 

Given a 2$S^{\left(2\right)}\left(1,3\right)$-design of order $v=3h$, there exists a 2$S^{\left(2\right)}\left(1,3\right)$-design of order $v^{\prime }=v+6=3h+6$.

Proof. 

Let us denote by ${\Sigma }_1=(X_1,{\mathcal{B}}_1)$ and ${\Sigma }_2=(X_2,{\mathcal{B}}_2)$ two block designs of type 2$S^{\left(2\right)}\left(1,3\right)$ of order $3h$ and 6, respectively. By Lemma 9, given $\alpha \in X_1$, one can construct a 2$S^{\left(2\right)}\left(1,3\right)$-design over $\left\{\alpha \right\}\cup X_2$ and let ${\mathcal{C}}_{\alpha }$ denote the set of blocks over $\alpha \cup X_2$. Analogously, for each $\beta \in X_2$, one can construct a 2$S^{\left(2\right)}\left(1,3\right)$-design over $X_1\cup \left\{\beta \right\}$ and denote by ${\mathcal{C}}_{\beta }$ the set of blocks over $X_1\cup \left\{\beta \right\}$. One can check that ${\mathcal{B}}_1\cup {\mathcal{B}}_2{\cup }_{\alpha \in X_1}{\mathcal{C}}_{\alpha }{\cup }_{\beta \in X_2}{\mathcal{C}}_{\beta }$ are the blocks of a 2$S^{\left(2\right)}\left(1,3\right)$-design over $X_1\cup X_2$.  □

Corollary 3. 

There exist 2$S^{\left(2\right)}\left(1,3\right)$-designs of order $3h$ if $h\ge 2$.

Proof. 

By Lemmas 7, 8 and 10, the conclusion is immediate.  □

Corollary 4. 

There exist 2$S^{\left(2\right)}\left(1,3\right)$-designs of order $v=3h+1$.

In order to prove the main result of this section, we also need the following lemmas.

Lemma 11. 

There exists a 2$S^{\left(3\right)}\left(2,3\right)$-design of order 9.

Proof. 

Take two disjoint families $(X,{\mathcal{C}}_1)$ and $(X,{\mathcal{C}}_2)$ of $P^{\left(3\right)}\left(2,4\right)$-designs over a set X of cardinality 8, as in Theorem 6, that are edge-balanced and consider $(X,{\mathcal{C}}_1\cup {\mathcal{C}}_2)$ that is edge-balanced. For each block $[(x,y),z,t]\in {\mathcal{C}}_1\cup {\mathcal{C}}_2$, consider the set $\mathcal{B}$ of blocks of the form $\left[\right(x,y),z,t,★]$; it is easy to check that $(X\cup \{★\},\mathcal{B})$ is a 2$S^{\left(3\right)}\left(2,3\right)$-design.  □

Lemma 12. 

There exist 2$S^{\left(3\right)}\left(2,3\right)$-designs of order v if $v\equiv 0mod9$ and $v\ge 9$.

Proof. 

The proof proceeds by induction, with a base case which holds true by Lemma 11. Let ${\Sigma }_1=(X_1,{\mathcal{B}}_1)$ and ${\Sigma }_2=(X_2,{\mathcal{B}}_2)$ be two 2$S^{\left(3\right)}\left(2,3\right)$ designs of order $v_1=9h$ and $v_2=9$. Let us denote by $(X_1,{\mathcal{C}}_1)$ a 2$S^{\left(2\right)}\left(1,3\right)$-design over $X_1$ and for every block $\gamma =[\left(x\right),y,z,t]\in {\mathcal{C}}_1$ and $a\in X_2$, consider ${\gamma }_a=[(a,x),y,z,t]$ and let ${\mathcal{F}}_1$ denote the set of blocks ${\gamma }_a$, i.e., ${\mathcal{F}}_1=\{{\gamma }_a:\gamma \in {\mathcal{C}}_1,a\in X_2\}$. Analogously, consider a 2$S^{\left(2\right)}\left(1,3\right)$-design $(X_2,{\mathcal{C}}_2)$ and ${\mathcal{F}}_2=\{{\gamma }_b:\gamma \in {\mathcal{C}}_2,b\in X_1\}$. The reader can easily check that $(X_1\cup X_2,{\mathcal{B}}_1\cup {\mathcal{B}}_2\cup {\mathcal{F}}_1\cup {\mathcal{F}}_2)$ is a 2$S^{\left(3\right)}\left(2,3\right)$-design of order $v^{\prime }=v+9=9h+9$.  □

Lemma 13. 

There exist 2$S^{\left(3\right)}\left(2,3\right)$-designs of order $v=9h+1$.

Proof. 

By Lemma 12, there exist 2$S^{\left(3\right)}\left(2,3\right)$-designs of order $9h$ which we will denote by $\Sigma =(X,\mathcal{B})$; furthermore, by Lemma 12, there exists an $(X,\mathcal{C})$ with blocks isomorphic to $S^{\left(2\right)}\left(1,3\right)$ and index 2. For each $\gamma \in \mathcal{C}$, consider ${\gamma }_{★}=[(★,x),y,z,t]$ and let $\mathcal{F}=\{{\gamma }_{★}:\gamma \in \mathcal{C}\}$; it is easy to verify that $(X\cup \{★\},\mathcal{B}\cup \mathcal{F})$ is a 2$S^{\left(3\right)}\left(2,3\right)$-design of order $v=9h+1$.  □

Lemma 14. 

There exist 2$S^{\left(3\right)}\left(2,3\right)$-designs of order v if $v\equiv 2mod9$.

Proof. 

By the preceding lemma, there exists a 2$S^{\left(3\right)}\left(2,3\right)$-design of order $9h+1$, and by Corollary 3, there exists a 2$S^{\left(2\right)}\left(1,3\right)$-design over X which we denote by $(X,\mathcal{C})$. Define $\mathcal{F}=\{{\gamma }_{★}:\gamma \in \mathcal{C}\}$; it is easy to check that $(X\cup \{★\},\mathcal{B}\cup \mathcal{F})$ is a 2$S^{\left(3\right)}\left(2,3\right)$-design of order $9h+2$.  □

Collecting together all the previous results, the following results.

Theorem 9. 

There exist 2$S^{\left(3\right)}\left(2,3\right)$-designs of order v if and only if

$$v\equiv 0\mathit{or}v\equiv 1\mathit{or}v\equiv 2mod9\mathit{and}v\ge 9$$

5. The Spectrum of ${\mathit{S}}^{\left(\mathbf{3}\right)}\left(\mathbf{2},\mathbf{3}\right)$-Designs of Index 3

In this section, we determine the spectrum of $S^{\left(3\right)}\left(2,3\right)$-designs with index 3 (for short 3$S^{\left(3\right)}\left(2,3\right)$-designs) with the further condition that in the system, all the blocks are distinct. We prove that v belongs to $\mathsf{Spectrum}\left(3S^{\left(3\right)}\left(2,3\right)\right)$ if and only if $v\ge 5$. We need some preliminary lemmas.

Lemma 15. 

There exist 3$S^{\left(2\right)}\left(1,3\right)$-designs of order v iff $v\ge 4$.

Proof. 

It is immediately clear that there exists a (unique) 3$S^{\left(2\right)}\left(1,3\right)$-design of order 4 and there are not any for $v\le 3$. 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 $i\in \{1,2,3\}$, $j\in {\mathbb{Z}}_v=\{1,\dots ,v\}$ 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 ${\mathbb{Z}}_v$ appears three times. Denote by $a_{•j}$ the j-th column of a and consider the graph $(★)a_{•j}$, which is isomorphic to $S^{\left(2\right)}\left(1,3\right)$. Denote by $\mathcal{F}$ the set of these graphs.

By induction, there exists a 3$S^{\left(2\right)}\left(1,3\right)$-design $(X,\mathcal{B})$ of order v; it is easy to check that $(X\cup \{★\},\mathcal{B}\cup \mathcal{F})$ is a 3$S^{\left(2\right)}\left(1,3\right)$-design of order $v+1$.  □

Using the previous results, we get the following.

Theorem 10. 

There exist 3$S^{\left(3\right)}\left(2,3\right)$-designs of order v iff $v\ge 5$.

Proof. 

We can proceed by induction with the base case $v=5$, where $(X,\mathcal{D})$ is an $S^{\left(3\right)}\left(2,3\right)$-design of order $v=5$ with $X={\mathbb{Z}}_5$, and $\mathcal{D}$ is the following set:

$$\begin{array}{cc}\hfill \{& \left[\right(1,2),3,4,5],\left[\right(2,4),1,3,5],\left[\right(2,3),4,5,1],\left[\right(3,5),2,4,1],\left[\right(3,4),5,1,2],\hfill \\ & \left[\right(4,1),3,5,2],\left[\right(4,5),1,2,3],\left[\right(5,2),4,1,3],\left[\right(5,1),2,3,4],\left[\right(1,3),5,2,4]\}.\hfill \end{array}$$

By the induction hypothesis, there exists a 3$S^{\left(3\right)}\left(2,3\right)$-design $(X,\mathcal{B})$ of order v, and by Lemma 15, there exists a 3$S^{\left(2\right)}\left(1,3\right)$-design $(X,\mathcal{C})$. For each $\gamma =\left[\right(x),y,z,t]\in \mathcal{C}$, consider ${\gamma }_{★}=[(★,x),y,z,t]$ and $\mathcal{F}=\{{\gamma }_{★}:\gamma \in \mathcal{C}\}$. It is easy to check that $(X\cup \{★\},\mathcal{B}\cup \mathcal{F})$ is a 3$S^{\left(3\right)}\left(2,3\right)$-design.  □

6. The Spectrum of ${\mathit{S}}^{\left(\mathbf{3}\right)}\left(\mathbf{2},\mathbf{3}\right)$-Designs of Index $\mathit{\lambda }>\mathbf{3}$

For the sake of completeness, in this section, we deal with the problem of existence of $\lambda$$S^{\left(3\right)}\left(2,3\right)$-designs for any index $\lambda$, where the blocks are repeated with multiplicity $\lambda$.

To be more precise, we prove that if $\lambda \equiv 0,1mod3$, then there exist $\lambda$$S^{\left(3\right)}\left(2,3\right)$-designs of order v if and only if $v\equiv 0,1,2mod9$, while if $\lambda \equiv 0mod3$, then there exist $\lambda$$S^{\left(3\right)}\left(2,3\right)$-designs for every $v\ge 5$.

It is an immediate consequence of Theorem 4 that if $\lambda \equiv 1,2mod3$ and a $\lambda$$S^{\left(3\right)}\left(2,3\right)$-design exists, then $v\equiv 0,1,2mod9$.

In Section 3, we have already proven that if $v\equiv 0,1,2mod9$, then there exists a $S^{\left(3\right)}\left(2,3\right)$-design $(X,\mathcal{B})$ of order v and index $\lambda =1$. If repetitions of blocks are allowed, to construct a $\lambda$$S^{\left(3\right)}\left(2,3\right)$-design of order v, it is sufficient to consider the design $(X,\lambda \mathcal{B})$, where $\lambda \mathcal{B}$ is the uniform multiset with underlying set $\mathcal{B}$ and multiplicity $\lambda$.

Similar arguments apply to the case $\lambda \equiv 0mod3$ and $v\ge 5$: in Section 5, we proved that there exists a 3$S^{\left(3\right)}\left(2,3\right)$-design $(X,\mathcal{B})$ of order $v\ge 5$; in order to construct a $\lambda$$S^{\left(3\right)}\left(2,3\right)$-design, it is sufficient to consider the design $(X,\frac{\lambda }{3}\mathcal{B})$, where $\frac{\lambda }{3}\mathcal{B}$ is the uniform multiset with underlying set $\mathcal{B}$ and multiplicity $\frac{\lambda }{3}$.

To conclude our paper, we suggest the following open problem:

Determining the spectrum of $S^{\left(3\right)}(2,n)$ for any positive integer n.