The Cycle Decomposition of Multiple Complete 3-Uniform Hypergraphs

Abstract

This paper investigates the decomposition of the $\lambda$-fold complete 3-uniform hypergraph $\lambda K_{\nu }^{\left(3\right)}$ into 4-cycles, denoted as $S_{\lambda }(3,{\mathrm{\Gamma }}_{5,1},v)$. Using the ${\mathrm{\Gamma }}_{5,1}$-structure as a model, we develop recursive construction techniques that exploit symmetric properties and provide explicit designs for small orders. These recursive frameworks enable the systematic generation of large-order hypergraph designs from smaller building blocks, illustrating the symmetric inheritance of structural properties. We establish that the necessary conditions for such a decomposition are also sufficient: an $S_{\lambda }(3,{\mathrm{\Gamma }}_{5,1},v)$ exists if and only if $24\mid \lambda v(v-1)(v-2),2\mid \lambda (v-1)(v-2),\mathrm{and}v\ge 5.$ This result highlights the deep interplay between combinatorial design theory and symmetry in hypergraph decompositions.

1. Introduction

A t-uniform hypergraph H is defined by a vertex set V and a collection E of t-element subsets of V, known as hyperedges. The order of H is given by $\left|V\right|$. While 2-uniform hypergraphs correspond to ordinary graphs, the case $t\ge 3$ introduces richer structural complexity. The hypergraph $\lambda K_v^{\left(t\right)}$ denotes the $\lambda$-fold complete t-uniform hypergraph on v vertices, where each t-subset appears exactly $\lambda$ times. When $\lambda =1$, the hypergraph is simple. Further information on hypergraphs can be found in [1].

A decomposition of a graph G into subgraphs $H_1,\dots ,H_k$—denoted $G=H_1\oplus \dots \oplus H_k$—occurs when the edge sets of the $H_i$ partition the edges of G. If each $H_i$ is isomorphic to a fixed graph H, we say G is H-decomposable. In particular, if H is an m-cycle $C_m$, the decomposition is called a cycle decomposition [2]. A 2-hypercycle is known as a cycle or Berge cycle which was introduced by Berge [3].

Extending these notions to hypergraphs, a decomposition of a t-uniform hypergraph $H=(V,E)$ is a partition of E into sub-hypergraphs $H_1,\dots ,H_k$ such that

• ${\bigcup }_{i=1}^{k}E\left(H_i\right)=E$;
• $E\left(H_i\right)\cap E\left(H_j\right)=Ø$ for $i\ne j$.

Let X be a set of v points and K a set of positive integers. A t-wise balanced design (t-BD) of order v and index $\lambda$ is a pair $(X,\mathcal{A})$, where X is a v-set and $\mathcal{A}$ is a family of blocks (subsets of X) such that every t-subset of X is contained in exactly $\lambda$ blocks. Such a design is denoted $S_{\lambda }(t,K,v)$, where K is the set of allowable block sizes. When $K=\left\{k\right\}$, we write $S_{\lambda }(t,k,v)$. Special cases include Steiner triple systems $S(2,3,v)$ and Steiner quadruple systems $S(3,4,v)$, whose existence conditions are classical. An $S(3,4,v)$ is called a Steiner quadruple system of order v, denoted by $SQS\left(v\right)$. An $SQS\left(v\right)$ exists if and only if $v\equiv 2,4(mod6)$ [4,5]. An $S(2,3,v)$ is called a Steiner triple system of order v and denoted by $STS\left(v\right)$. There exists an $STS\left(v\right)$ if and only if $v\equiv 1,3(mod6)$ [6].

A t-hypercycle decomposition of a t-uniform hypergraph H refers to a partition of its edge set into hypercycles. This concept naturally extends the classical notion of cycle decomposition from graphs to hypergraphs. For the graph case ($t=2$), the problem of decomposing $\lambda K_n^{\left(2\right)}$ into cycles of length m has been fully resolved for $\lambda =1,2$ and $m\ge 2$; see references [7,8].

Let $\mathrm{\Gamma }$ be a collection of specific 3-uniform hypergraphs (particularly 4-cycles) that we aim to use in the decomposition of $\lambda k_V^{\left(3\right)}$. In this work, we focus on decomposing $\lambda K_v^{\left(3\right)}$ into 4-cycles of a specific type ${\mathrm{\Gamma }}_{5,1}$, leveraging tools from group-divisible designs and candelabra systems to construct such decompositions recursively.

A group-divisible t-design of order v with block sizes from a set K, denoted $\mathrm{GDD}(t,K,v)$, is defined as a triple $(X,\mathcal{G},\mathcal{B})$, satisfying the following conditions [9]:

• X is a set of v called points;
• $\mathcal{G}$ forms a partition of X into nonempty subsets called groups (or holes);
• $\mathcal{B}$ consists of blocks, each being a k-subset of X (with $k\in K$) that intersects every group in at most one point (such a subset is called a k-transverse);
• Every k-transverse of $\mathcal{G}$ is contained in exactly one block.

The type of a $\mathrm{GDD}$ is the multiset of group sizes. If there are $n_i$ groups of size $g_i$ for $1\le i\le r$, the type is denoted by $g_1^{n_1}{g}_2^{n_2}\dots g_r^{n_r}$. A $\mathrm{GDD}$ is said to be uniform if all groups have the same size. Note that a $\mathrm{GDD}(t,K,v)$ of type $1^v$ corresponds to an $S(t,K,v)$ design. When $K=k$, we simplify the notation to $\mathrm{GDD}(t,k,v)$.

While group-divisible 2-designs have been thoroughly studied [10], we concentrate here on the case $t=3$ and block size four. A $\mathrm{GDD}(3,4,v)$ is also known as a transverse Steiner quadruple system [11,12]. A uniform $\mathrm{GDD}(3,4,mg)$ of type $g^m$ is referred to as an $H(m,g,4,3)$ design. The existence spectrum for such H-designs has been established by Mills [13] and Ji [14,15].

Let $v,m,t$ be positive integers, and s be a non-negative integer. Let X be a set of $v=s+{\Sigma }_{i=1}^m{a}_i{g}_i$ points, S be a partition of V into subsets of size s, called a stem, and $\mathcal{G}=\{G_1,G_2\dots \}$ is a division of the $X\setminus S$. The elements in $\mathcal{G}$ are called groups; $\mathrm{\Gamma }$ is the vertex set defined in families of supergraphs on some subsets on X. Each hypergraph in $\mathcal{B}$ is isomorphic to a hypergraph in $\mathrm{\Gamma }$, and each hypergraph in $\mathrm{\Gamma }$ is called a block. A $(\mathrm{\Gamma },t)$ is represented by an ordered quadruple $(X,S,\mathcal{G},\mathrm{\Gamma })$, where for any t-subset T of X, if $|T\cap (S\cup G_i\left)\right|<t$ for each i are established, then T is in exactly one block of $\mathcal{G}$; otherwise, any t-subset of $S\cup G_i$ is not contained in any block of $\mathcal{G}$. This candelabra system is denoted as $\mathrm{CS}(s,\mathrm{\Gamma },v)$.

From the definition of the $Candelabra$ $Design$, it follows that if $(X,\mathcal{B})$ is an $S(t,K,v)$ with $t\ge 3$, then for any $x_0\in X$, the quadruple $(X,\left\{x_0\right\},\{\left\{x\right\}:x\in X\setminus \left\{x_0\right\}\},\mathcal{B})$, is a $\mathrm{CS}(t,K,v)$-design of type $(1^{v-1}:1)$.

According to the definition of the Candelabra $(\mathrm{\Gamma },t)$-design, we know that if $(X,\mathcal{B})$ is an $S_{\lambda }(t,\mathrm{\Gamma },v)$, where $t\ge 3$, then for any $x_0\in X$, the quadruple $(X,Ø,\{\left\{x\right\}:x\in X\setminus \left\{x_0\right\}\},\mathcal{B})$ is a ${\mathrm{CS}}_{\lambda }(t,\mathrm{\Gamma },v)$-design of type $(1^v:0)$. In the following construction, $\mathrm{CS}$ stands for Candelabra System.

Table 1. Types of 4-cycles defined on the set $\{1,2,\dots ,v\}$.

v	Notation	Type of 4-Cycles
4	$K_4^3$	$\{1,3,2\},\{2,4,3\},\{3,1,4\},\{4,2,1\}$
5	$W_4^3$	$\{1,5,2\},\{2,5,3\},\{3,5,4\},\{4,5,1\}$
	${\mathrm{\Gamma }}_{5,1}$	$\{1,5,2\},\{2,5,3\},\{3,2,4\},\{4,5,1\}$
	${\mathrm{\Gamma }}_{5,2}$	$\{1,5,2\},\{2,4,3\},\{3,1,4\},\{4,2,1\}$
	${\mathrm{\Gamma }}_{5,3}$	$\{1,5,2\},\{2,1,3\},\{3,2,4\},\{4,2,1\}$
	${\mathrm{\Gamma }}_{5,4}$	$\{1,5,2\},\{2,1,3\},\{3,5,4\},\{4,2,1\}$
6	$P_4^3$	$\{1,5,2\},\{2,6,3\},\{3,5,4\},\{4,6,1\}$
	${\mathrm{\Gamma }}_{6,1}$	$\{1,5,2\},\{2,4,3\},\{3,5,4\},\{4,6,1\}$
	${\mathrm{\Gamma }}_{6,2}$	$\{1,5,2\},\{2,4,3\},\{3,6,4\},\{4,6,1\}$
	${\mathrm{\Gamma }}_{6,3}$	$\{1,5,2\},\{2,1,3\},\{3,1,4\},\{4,6,1\}$
	${\mathrm{\Gamma }}_{6,4}$	$\{1,5,2\},\{2,4,3\},\{3,1,4\},\{4,6,1\}$
	${\mathrm{\Gamma }}_{6,5}$	$\{1,5,2\},\{2,5,3\},\{3,1,4\},\{4,6,1\}$
	${\mathrm{\Gamma }}_{6,6}$	$\{1,5,2\},\{2,5,3\},\{3,6,4\},\{4,6,1\}$
7	${\mathrm{\Gamma }}_{7,1}$	$\{1,5,2\},\{2,6,3\},\{3,7,4\},\{4,2,1\}$
	${\mathrm{\Gamma }}_{7,2}$	$\{1,5,2\},\{2,6,3\},\{3,7,4\},\{4,5,1\}$
	${\mathrm{\Gamma }}_{7,3}$	$\{1,5,2\},\{2,6,3\},\{3,7,4\},\{4,6,1\}$
8	$L_4^3$	$\{1,5,2\},\{2,6,3\},\{3,7,4\},\{4,8,1\}$

provides the types of 4-cycles under isomorphism.

According to Table 1, ${\mathrm{\Gamma }}_{5,1}=\left\{\left\{v_1,v_5,v_2\right\},\left\{v_2,v_5,v_3\right\},\left\{v_3,v_2,v_4\right\},\left\{v_4,v_5,v_1\right\}\right\}$, denoted as $\{v_1,v_2,v_3,v_4,v_5\}$.

The hypergraph of ${\mathrm{\Gamma }}_{5,1}$ is taken as an example, as shown in Figure 1 below.

Figure 1 illustrates the structure of the ${\mathrm{\Gamma }}_{5,1}$ hypergraph, containing the vertex set $1,2,3,4,5$. The four hyperedges are represented in different colors: $1,5,2$ (blue), $2,5,3$ (orange), $3,2,4$ (green), and $4,5,1$ (yellow). Thick lines are used to connect vertices within the same hyperedge to visually display their composition.

Lemma 1 

([16]). Let Γ be a collection of 3-uniform hypergraphs, and let $J\in \mathrm{\Gamma }$. Let $d_J\left(x\right)$ be the number of hyperedges in J containing vertex x, and let $d_J^{*}(x_1,x_2)$ be the number of hyperedges in J containing both $x_1$ and $x_2$. The necessary conditions for the existence of an $S(3,\mathrm{\Gamma },v)$-design are as follows:

$\left(1\right)$ $v\ge min\left\{\right|V\left(J\right)|:J\in \mathrm{\Gamma }\}$;

$\left(2\right)$ $\left({\displaystyle \genfrac{}{}{0pt}{}{v}{3}}\right)\equiv 0\left(\mathit{mod}{d}_0\right),$ where $d_0=gcd\left\{\left|E\right(J\left)\right|:J\in \mathrm{\Gamma }\right\};$

$\left(3\right)$ $\left({\displaystyle \genfrac{}{}{0pt}{}{v-1}{2}}\right)\equiv 0\left(\mathit{mod}{d}_1\right)$ where $d_1=gcd\left\{d_J\left(x\right):x\in V\left(J\right),J\in \mathrm{\Gamma }\right\};$

$\left(4\right)$ $\left({\displaystyle \genfrac{}{}{0pt}{}{v-2}{1}}\right)\equiv 0\left(\mathit{mod}{d}_2\right),$ where $d_2=gcd\left\{d_J^{*}\left(x_1,x_2\right):x_1\ne x_2,x_1,x_2\in \right.$$V\left(J\right),J\in \mathrm{\Gamma }\}.$

Lemma 2 

([2]). When $m=4,6$, $g=2,3,$ and $(m,g)\ne (6,2)$ a $H(m,g,4,3)$-design exists.

Lemma 3 

([8]). The necessary and sufficient condition for the existence of a $H(m,g,4,3)$-design is $m\ge 4$, $mg\equiv 0$$\left(\mathit{mod}2\right)$, $(m-1)(m-2)\equiv 0\left(\mathit{mod}3\right)$, except for $(m,g)=(5,2)$.

Lemma 4 

([8]). For $m\equiv 1,2\left(\mathit{mod}3\right)$ and $m\ge 4$, a $H(m,4,4,3)$-design exists.

In this study, 2-$FG(3,(3,3,4,6),2k)$ denotes a frame group-divisible design with index 2, block sizes from the set $\{4,6\}$, used for decomposing specific hypergraph structures, of type $2^k$ (or $2^{k-2}{4}^1$). For detailed definitions, see reference [6].

Lemma 5. 

$\left(1\right)$ For any $k\equiv 2\left(\mathit{mod}3\right)$ and $k\ge 5$, there exists a 2-$FG(3,(3,3,\{4,6\}),2k)$-design of type $2^{k-2}{4}^1$.

$\left(2\right)$ For any $k\equiv 0,1\left(\mathit{mod}3\right)$ and $k\ge 5$, there exists a type 2-$FG(3,(3,3,\{4,6\}),2k)$-design of type $2^k$.

Let $s\le v$; for convenience, the design $(K_v^t\setminus K_s^t$, $\mathrm{\Gamma })$ is recorded as $HS(t,\mathrm{\Gamma };v,s)$. The set of points among s is called the hole of this design.

Based on the lemma above, the following corollary can be proposed.

Corollary 1. 

The necessary conditions for the existence of an $S_{\lambda }(3,\mathrm{\Gamma },v)$-design are

$$24\left|\lambda v\right(v-1\left)\right(v-2),2|\lambda (v-1)(v-2),$$

and $v\ge 5$.

When $\lambda$ and v take different values, the sufficiency of $S_{\lambda }(3,\mathrm{\Gamma },v)$-design can be discussed. The necessary conditions for the existence of $S_{\lambda }(3,\mathrm{\Gamma },v)$-design can be divided and organized into the following three cases:

$\left(1\right)$ $\lambda \equiv 1,3(mod4)$, $v\equiv 0,2,4(mod6)$ and $v\equiv 1(mod8)$ with $v\ge 6$;

$\left(2\right)$ $\lambda \equiv 2(mod2)$, $v\equiv 0,1,2,4,5,6,8,9,10(mod12)$ with $v\ge 5$;

$\left(3\right)$ $\lambda \equiv 0(mod4)$, with $v\ge 5$.

2. Recursive Constructions

Construction 1. 

(1) Assume a group-divisible design $GDD(t,K,v)$ of type $g_1^{a_1}{g}_2^{a_2}\dots g_m^{a_m}$ is given.(2) If for every $k\in K$ there exists a $GDD(t,\mathrm{\Gamma },hk)$ of type $h^k$, then a $GDD(t,\mathrm{\Gamma },hv)$ of type ${\left(h{g}_1\right)}^{a_1}{\left(h{g}_2\right)}^{a_2}\dots {\left(h{g}_m\right)}^{a_m}$ exists.

Proof of Construction 1. 

Let $(X,\mathcal{G},\mathcal{B})$ be a $GDD(t,K,v)$ of type $g_1^{a_1}{g}_2^{a_2}\dots g_m^{a_m}$, where $\mathcal{G}=\{G_1,G_2,\dots ,G_n\}$ and $n={\sum }_{i=1}^m{a}_i{g}_i$.

For any $B\in \mathcal{B}$, on the point set $V\left(B\right)\times I_h$, the group formed by $\left\{x\right\}\times I_h$, $x\in V\left(B\right)$, a $GDD(t,\mathrm{\Gamma },hk)$ of type $h^k$, and its block set is denoted as ${\mathcal{A}}_B$.

Let $X^{\prime }=X\times I_h$, ${\mathcal{G}}^{\prime }=G_i\times I_h:1\le i\le n.$ Let $\mathcal{A}={\cup }_{B\in \mathcal{B}}{\mathcal{A}}_B$, then $(X^{\prime },{\mathcal{G}}^{\prime },\mathcal{A})$ is a $GDD(t,\mathrm{\Gamma },hv)$.

Therefore, the conclusion holds.

This completes the proof. □

Construction 2. 

Assume there exists a $GD{D}_{\lambda }(3,\mathrm{\Gamma },gn)$-design of type $g^n$; if there exists a $C{S}_{\lambda }(3,\mathrm{\Gamma },2g+s)$-design of type $\left(g^2:s\right)$, then a $C{S}_{\lambda }(3,\mathrm{\Gamma },gn+s)$-design of type $\left(g^n:s\right)$ exists.

Proof of Construction 2. 

Let $(X,\mathcal{G},\mathcal{B})$ be a ${\mathrm{GDD}}_{\lambda }(3,\mathrm{\Gamma },v)$-design of type $g^n$, where $\mathcal{G}=\left\{G_1,G_2,\dots ,G_n\right\}$. For any $1\le i<j\le n$, construct a $C{S}_{\lambda }(3,\mathrm{\Gamma },2g+s)$-design of type $(g^2:s)$ on $G_i\cup G_j\cup S$ with stem S and groups $G_i$ and $G_j$, and denote its block set as ${\mathcal{A}}_{i,j}$. Let $\mathcal{A}={\cup }_{1\le i<j\le n}{\mathcal{A}}_{i,j}$, then $(X,\mathcal{G},\mathcal{B},\mathcal{A})$ is a ${\mathrm{CS}}_{\lambda }(3,\mathrm{\Gamma },gn+s)$-design of type $\left(g^n:s\right)$.

This completes the proof. □

Construction 3. 

If there exists a $C{S}_{\lambda }(t,\mathrm{\Gamma },v)$-design of type $\left(g_1^{a_1}{g}_2^{a_2}\dots g_{m-1}^{a_{m-1}}{g}_m^1:s\right)$, where $v=$${\sum }_{i=1}^{m-1}{a}_i{g}_i+g_m+s$, if

(1)

For any $1\le i\le m-1$, there exists a $H{S}_{\lambda }\left(t,\mathrm{\Gamma };g_i+s,s\right)$;

(2)

And there exists an $S_{\lambda }\left(t,\mathrm{\Gamma },g_m+s\right)$, then an $S_{\lambda }(t,\mathrm{\Gamma },v)$-design exists.

Proof of Construction 3. 

Let $(X,\mathcal{S},\mathcal{G},\mathcal{B})$ be a ${\mathrm{CS}}_{\lambda }(t,\mathrm{\Gamma },v)$-design of type $\left(g_1^{a_1}{g}_2^{a_2}\dots g_{m-1}^{a_{m-1}}{g}_m^1:s\right)$, for each group $G\in \mathcal{G}$, and $1\le i\le m-1$ of size $g_i$; then construct an $H{S}_{\lambda }\left(t,\mathrm{\Gamma };g_i+\right.$$s,s)$-design with S as the hole on $G\cup S$, and denote its block set as ${\mathcal{A}}_i$. For a group G of size $g_m$, construct an $S_{\lambda }\left(t,\mathrm{\Gamma },g_m+s\right)$-design on $G\cup S$, and denote its block set as ${\mathcal{A}}_m$. Let $\mathcal{A}=U_{1\le i\le m}{\mathcal{A}}_i$, then $(\mathcal{X},\mathcal{A})$ is an $S_{\lambda }(3,\mathrm{\Gamma },v)$.

This completes the proof. □

Construction 4. 

Let $(X,S,\mathcal{G},\mathcal{A})$ be a $CS(3,\mathrm{\Gamma },v)$-design of type $\left(g_1^{a_1}{g}_2^{a_2}\dots g_{m-1}^{a_{m-1}}{g}_m^{a_m}:s\right)$, where $S=\left\{x_1,x_2,\dots ,x_s\right\}$, $v={\sum }_{i=1}^m{a}_i{g}_i+s$. If

(1) 

For each block A containing $x_1$, there exists a $CS(3,\mathrm{\Gamma },b(\left|V\left(A\right)\right|-1)+b_1)$-design of type $\left(b^{\left|V\right(A\left)\right|-1}:b_1\right)$;

(2) 

For each block A containing $x_i(2\le i\le s)$, there exists a $GDD\left(3,\mathrm{\Gamma },b\left(\right|V\left(A\right)|-1)+b_i\right)$-design of type $\left(b^{\left|V\right(A\left)\right|-1}{b}_i\right)$;

(3) 

For each block A not containing $x_i(1\le i\le s)$, there exists a $\mathrm{GDD}(3,\mathrm{\Gamma },b(\left|V\right(A\left)\right|\left)\right)$-design of type $b^{\left|V\right(A\left)\right|}$, then there exists a $CS(3,\mathrm{\Gamma },v^{\prime })$-design of type $({\left({\mathrm{bg}}_1\right)}^{a_1}{\left({\mathrm{bg}}_2\right)}^{a_2}\dots {\left({\mathrm{bg}}_{\mathrm{m}}\right)}^{a_m}:{\sum }_{i=1}^s{b}_i)$, where $v^{\prime }=(v-s)b+{\sum }_{i=1}^s{b}_i$.

Proof of Construction 4 

For each block $A\in \mathcal{A}$, when $x_1\in A$, on the point set $\left(V\left(A\right)\setminus \left\{x_1\right\}\times I_b\right)\cup S^{\prime }$, construct a $CS(3,\mathrm{\Gamma },b(\left|V\left(A\right)\right|-1)+b_1)$-design of type $(b^{\left|V\right(A\left)\right|-1}:b_1)$, whose group set is $\left\{\left\{x\right\}\times I_b:x\in V\left(A\right)\setminus \left\{x_1\right\}\right\}$, where $S^{\prime }$ is its stem, and when $\left|S^{\prime }\right|=b_1$, denote its block set as ${\mathcal{B}}_{A\left(x_1\right)}$.

When $x_i\in A$, $2\le i\le s$, construct a $\mathrm{GDD}\left(3,\mathrm{\Gamma },b\left(\right|V\left(A\right)|-1)+b_i\right)$-design with type $b^{\left|V\right(A\left)\right|-1}{b}_i$, whose group set is $\left\{\right\{x\}\times$$\left.I_b+I_{b_i}:x\in V\left(A\right)\right\}$ on the point set $(V\left(A\right)-1)\times I_b+I_{b_i}$, and denote its block set as ${\mathcal{B}}_{A\left(x_i\right)}$.

When for any $1\le i\le s$, $x_i\notin A$, construct a $\mathrm{GDD}(3,\mathrm{\Gamma },b|V\left(A\right)\left|\right)$-design with type $b^{\left|V\right(A\left)\right|}$, whose group set is $\left\{\right\{x\}\times$$\left.I_b:x\in V\left(A\right)\right\}$ on the point set $V\left(A\right)\times I_b$, and denote its block ${\mathcal{B}}_A$.

Let $X^{\prime }=\left(\left(X\setminus \left\{x_1\right\}\right)\times I_b\right)\cup S^{\prime },{\mathcal{G}}^{\prime }=\left\{G\times I_b:G\in \mathcal{G}\right\}$, and $S^{″}=\left({\cup }_{2\le i\le s}\left(\left\{x_i\right\}\times \right.\right.$$\left.\left.I_b\right)\right)\cup S^{\prime }$. Let ${\mathcal{B}}^{\prime }=U_{1\le i\le s}{\mathcal{B}}_{A\left(x_i\right)},{\mathcal{B}}^{″}=\left\{{\mathcal{B}}_A:x_i\in A,1\le i\le s,A\in \mathcal{A}\right\}$. Let $\mathcal{B}={\mathcal{B}}^{\prime }\cup {\mathcal{B}}^{″}$. It can be verified that $\left(X^{\prime },S^{″},{\mathcal{G}}^{\prime },\mathcal{B}\right)$ is a $CS\left(3,\mathrm{\Gamma },v^{\prime }\right)$-design.

This completes the proof. □

3. Some Small Orders

Lemma 6. 

For $v\in \{6,8,9,10,12\}$, there exists $S\left(3,{\mathrm{\Gamma }}_{5,1},v\right),S_3\left(3,{\mathrm{\Gamma }}_{5,1},v\right)$.

Proof. 

When $v=6$, let $V\left(K_6^{\left(3\right)}\right)=Z_5\cup \{\infty \}$. The base blocks are listed below. All other blocks are obtained by developing these base blocks by $+1$ modulo 5, where $\infty +1=\infty$.

$$(\infty ,1,4,2,0).$$

When $v=8$, let $V\left(K_8^{\left(3\right)}\right)=Z_7\cup \{\infty \}$. The base blocks are listed below. All other blocks are obtained by developing these base blocks by $+1$ modulo 7, where $\infty +1=\infty$.

$$(2,0,3,5,1),(1,0,4,6,\infty ).$$

When $v=9$, let $V\left(K_9^{\left(3\right)}\right)=Z_7\cup \left\{{\infty }_1,{\infty }_2\right\}$. The base blocks are listed below. All other blocks are obtained by developing these base blocks by $+1$ modulo 10, where ${\infty }_i+1={\infty }_i,i=1,2$.

$$(2,0,3,5,1),\left(2,0,{\infty }_1,{\infty }_2,3\right),\left({\infty }_1,2,{\infty }_2,6,0\right).$$

When $v=10$, let $V\left(K_{10}^{\left(3\right)}\right)=Z_{10}$. The base blocks are listed below. All other blocks are obtained by developing these base blocks by $+1$ modulo 10.

$$(2,0,3,5,1),(0,5,3,8,1),(6,1,7,3,0).$$

When $v=12$, let $V\left(K_{12}^{\left(3\right)}\right)=Z_{11}\cup \{\infty \}$. The base blocks are listed below. All other blocks are obtained by developing these base blocks by $+1$ modulo 11, where $\infty +1=\infty$.

$$(1,0,9,5,\infty ),(1,5,\infty ,3,0),(7,6,8,0,1),(0,8,5,2,1),(4,1,10,7,3).$$

When $v=\{6,8,9,10,12\}$, there exists a $S\left(3,{\mathrm{\Gamma }}_{5,1},v\right)$. Repeating the blocks of $S\left(3,{\mathrm{\Gamma }}_{5,1},v\right)$ three times yields $S_3\left(3,{\mathrm{\Gamma }}_{5,1},6\right)$. □

Lemma 7. 

For $v\in \{5,6,8,9,10,12,13,17\}$, there exists $S_2\left(3,{\mathrm{\Gamma }}_{5,1},v\right)$.

Proof. 

When $v=5$, let $V\left(K_5^{\left(3\right)}\right)=Z_5$. The base blocks are listed below. All other blocks are obtained by developing these base blocks by $+1$ modulo 5.

$$(0,4,2,3,1).$$

When $v=\{6,8,9,10,12\}$, there exists a $S\left(3,{\mathrm{\Gamma }}_{5,1},v\right)$. Repeating the blocks of $S\left(3,{\mathrm{\Gamma }}_{5,1},v\right)$ twice yields $S_2\left(3,{\mathrm{\Gamma }}_{5,1},6\right)$.

When $v=13$, let $V\left(K_{13}^{\left(3\right)}\right)=Z_{13}$. The base blocks are listed below. All other blocks are obtained by developing these base blocks by $+1$ modulo 13.

$$\begin{array}{cc}& (0,5,3,8,1),(6,1,7,3,0),(0,2,12,6,4),(0,5,1,8,4),(12,7,9,1,2),\hfill \\ & (0,5,11,10,1),(0,9,5,12,1),(0,1,9,8,3),(0,4,6,7,1),(2,6,3,1,0),\hfill \\ & (8,3,0,12,1).\hfill \end{array}$$

When $v=17$, since $17=1(mod8)$, there exists an $S\left(3,{\mathrm{\Gamma }}_{5,1},17\right)$. Repeating the blocks of $S\left(3,{\mathrm{\Gamma }}_{5,1},v\right)$ twice yields $S_2\left(3,{\mathrm{\Gamma }}_{5,1},17\right)$, and thus, ${\mathrm{CS}}_2\left(3,{\mathrm{\Gamma }}_{5,1},17\right)$ certainly exists. □

Lemma 8. 

There exists a $GD{D}_2\left(3,{\mathrm{\Gamma }}_{5,1},8\right)$ of type $4^2$.

Proof. 

Let $V\left(K_{4,4}^{\left(3\right)}\right)=\{0,2,4,6\}\cup \{1,3,5,7\}$; the base blocks are listed below. All other blocks are obtained by developing these base blocks by $+1$ modulo 8.

$$(0,1,6,5,2),(3,0,7,4,2),(3,0,5,6,2).$$

□

Lemma 9. 

There exists a $GD{D}_2\left(3,{\mathrm{\Gamma }}_{5,1},24\right)$ of type ${12}^2$.

Proof. 

Let $\lambda =2,g_1=4,g_2=4,h_1=3,h_2=3,a_1=2,a_2=2$; by Construction 1 and Lemma 8, there exists a ${\mathrm{GDD}}_2\left(3,{\mathrm{\Gamma }}_{5,1},24\right)$ of type ${12}^2$. □

Lemma 10. 

There exists a $GD{D}_2\left(3,{\mathrm{\Gamma }}_{5,1},8\right)$ of type $2^4$.

Proof. 

Let $V\left(K_{2,2,2,2}^{\left(3\right)}\right)=\{0,4\}\cup \{1,5\}\cup \{2,6\}\cup \{3,7\}$; the base blocks are listed below. All other blocks are obtained by developing these base blocks by $+1$ modulo 8.

$$(0,2,4,3,1),(2,3,6,5,0).$$

□

Lemma 11. 

There exists a $GD{D}_2\left(3,{\mathrm{\Gamma }}_{5,1},48\right)$ of type ${12}^4$.

Proof. 

Let $\lambda =2,g_i=2,a_i=4,h_i=6,i=1,2,3,4$; by Construction 1 and Lemma 10, there exists a ${\mathrm{GDD}}_2\left(3,{\mathrm{\Gamma }}_{5,1},48\right)$ of type ${12}^4$. □

Lemma 12.

There exists an $S_2\left(3,{\mathrm{\Gamma }}_{5,1},21\right)$.

Proof. 

When $v=21$, let $V\left(K_{21}^{\left(3\right)}\right)=Z_{19}\cup \left\{{\infty }_1,{\infty }_2\right\}$, the base blocks are listed below. All other blocks are obtained by developing these base blocks by $+1$ modulo 19, where ${\infty }_i+1={\infty }_i,i=1,2$.

$$\begin{array}{cc}& (0,2,4,3,1),(13,1,4,10,0),(2,1,0,5,6),(0,8,2,7,1),(0,1,2,10,9),\hfill \\ & (0,1,10,9,11),(6,7,8,5,0),(0,8,3,5,7),(6,2,7,10,4),(5,9,1,11,3),\hfill \\ & (4,9,0,11,2),(7,2,11,9,0),(0,8,1,4,6),(3,8,2,10,6),(5,3,8,16,0),\hfill \\ & (12,9,6,16,0),(11,3,12,7,0),(12,4,15,8,1),(0,4,17,10,7),(1,4,0,5,3),\hfill \\ & \left(0,8,14,{\infty }_1,4\right),(9,4,10,15,0),\left(4,0,5,{\infty }_2,10\right),\left(11,4,13,{\infty }_1,0\right),\hfill \\ & \left(4,0,6,{\infty }_2,14\right),\left(9,5,12,{\infty }_2,0\right),\left({\infty }_2,0,6,12,{\infty }_1\right),\left(7,0,5,10,{\infty }_1\right),\hfill \\ & \left(7,0,5,11,{\infty }_2\right),\left(3,0,5,12,{\infty }_1\right),\left({\infty }_1,1,{\infty }_2,2,0\right),\left({\infty }_2,0,{\infty }_1,18,2\right),\hfill \\ & \left(8,0,3,16,{\infty }_2\right),\left(1,0,{\infty }_2,17,4\right),\left(8,0,6,1,{\infty }_1\right).\hfill \end{array}$$

□

Lemma 13. 

There exists $GD{D}_2\left(3,{\mathrm{\Gamma }}_{5,1},10\right)$ and $GD{D}_4\left(3,{\mathrm{\Gamma }}_{5,1},10\right)$, both of type $2^3{4}^1$.

Proof. 

From the literature [17], it is known that there exists a $\mathrm{GDD}\left(3,{\mathrm{\Gamma }}_{5,1},2^3{4}^1\right)$-design of type $2^3{4}^1$. Therefore, by repeating the blocks of $GDD\left(3,{\mathrm{\Gamma }}_{5,1},2^3{4}^1\right)$ twice, we obtain ${\mathrm{GDD}}_2\left(3,{\mathrm{\Gamma }}_{5,1},2^3{4}^1\right)$, and by repeating them 4 times, we obtain ${\mathrm{GDD}}_4\left(3,{\mathrm{\Gamma }}_{5,1},2^3{4}^1\right)$. □

Lemma 14. 

There exists an $S_4\left(3,{\mathrm{\Gamma }}_{5,1},7\right)$.

Proof. 

When $v=7$, let $V\left(K_7^{\left(3\right)}\right)=Z_7$. The base blocks are listed below. All other blocks are obtained by developing these base blocks by $+1$ modulo 7.

$$(0,1,6,4,2),(2,1,3,5,0),(3,1,4,2,0),(1,4,6,5,0),(0,2,5,1,6).$$

□

Lemma 15. 

There exists an $S_4\left(3,{\mathrm{\Gamma }}_{5,1},11\right)$.

Proof. 

When $v=11$, let $V\left(K_{11}^{\left(3\right)}\right)=Z_{11}$. The base blocks are listed below. All other blocks are obtained by developing these base blocks by $+1$ modulo 11.

$$\begin{array}{cc}& (0,1,6,4,2),(2,1,3,5,0),(3,1,4,2,0),(1,4,6,5,0),(0,2,5,1,6),(3,1,4,6,0),\hfill \\ & (7,1,6,5,0),(7,1,6,3,0),(7,1,8,3,0),(9,1,8,3,0),(1,0,2,5,9),(4,6,3,8,0),\hfill \\ & (8,0,6,3,2),(1,6,0,3,2),(9,1,8,7,0).\hfill \end{array}$$

□

4. Conclusions

Theorem 1. 

If $\lambda \equiv 1,3\left(\mathit{mod}4\right)$, $v\equiv 0,2,4\left(\mathit{mod}6\right)$, and $v\equiv 1\left(\mathit{mod}8\right)$ with $v\ge 6$, then there exists an $S_{\lambda }\left(3,{\mathrm{\Gamma }}_{5,1},v\right)$.

Proof of Theorem 1. 

Prior results in [17] establish the existence of an $S\left(3,{\mathrm{\Gamma }}_{5,1},v\right)$-design under the stated conditions. For any positive integer $\lambda$, repeating each block of the $S\left(3,{\mathrm{\Gamma }}_{5,1},v\right)$-design $\lambda$ times yields an $S_{\lambda }\left(3,{\mathrm{\Gamma }}_{5,1},v\right)$-design.

This completes the proof. □

Theorem 2. 

If $\lambda \equiv 2\left(\mathit{mod}4\right)$, $v\equiv 0,1,2,4,5,6,8,9,10\left(\mathit{mod}12\right)$ with $v\ge 5$, then there exists an $S_{\lambda }\left(3,{\mathrm{\Gamma }}_{5,1},v\right)$.

Proof of Theorem 2. 

Let k be a positive integer. For $a\le 2$, it is known that $K_{12+a}^{\left(3\right)}\setminus K_a^{\left(3\right)}$ is isomorphic to $K_{12+a}^{\left(3\right)}$. To find the existence of $H{S}_2\left(3,{\mathrm{\Gamma }}_{5,1},12+a,a\right)$, it suffices to find $S_2\left(3,{\mathrm{\Gamma }}_{5,1},12+a\right)$. The proof of the existence of $S_{\lambda }\left(3,{\mathrm{\Gamma }}_{5,1},v\right)$-design is divided into the following cases:

$\left(1\right)$ By Lemma 7, $S_2\left(3,{\mathrm{\Gamma }}_{5,1},5\right)$, $S_2\left(3,{\mathrm{\Gamma }}_{5,1},9\right)$ exists.

$\left(2\right)$ By Lemma 6, and ref. [17], for $v\equiv 0,2,4,6,8,10(mod12)$, an $S\left(3,{\mathrm{\Gamma }}_{5,1},v\right)$-design exists. For any positive integer $\lambda$, repeating each block of the $S\left(3,{\mathrm{\Gamma }}_{5,1},v\right)$ -design $\lambda$ times yields an $S_{\lambda }\left(3,{\mathrm{\Gamma }}_{5,1},v\right)$-design.

$\left(3\right)$ For $v\equiv 1(mod12)$ and $v\ge 13$. By decomposition, the set $\{v\mid v\equiv 1(mod12\left)\right\}=\{v\mid v\equiv 1(mod8\left)\right\}\cup \{v\mid v\equiv 5(mod8\left)\right\}$.

When $v\equiv 1(mod8)$, an $S\left(3,{\mathrm{\Gamma }}_{5,1},v\right)$-design exists. For any positive integer $\lambda$, repeating each block of the $S\left(3,{\mathrm{\Gamma }}_{5,1},v\right)$-design $\lambda$ times yields an $S_{\lambda }\left(3,{\mathrm{\Gamma }}_{5,1},v\right)$-design.

When $v\equiv 5(mod8)$ and $v\ge 13$, let $v=8m+5$, where $m\equiv 1(mod3)$ and $m\ge 1$. By Lemma 4, a $H_2(m,4,4,3)$-design exists. Using Construction $2.1$, we can obtain a ${\mathrm{GDD}}_2\left(3,{\mathrm{\Gamma }}_{5,1},8m\right)$, and the required ${\mathrm{GDD}}_2\left(3,{\mathrm{\Gamma }}_{5,1},2^4\right)$-design is obtained from Lemma 10. By Lemma 8, an $S_2\left(3,{\mathrm{\Gamma }}_{5,1},21\right)$-design exists; thus a ${\mathrm{CS}}_2\left(3,{\mathrm{\Gamma }}_{5,1},21\right)$-design of type $\left(8^2:5\right)$ certainly exists, and using Construction 2, we can obtain a $CS\left(3,{\mathrm{\Gamma }}_{5,1},8m+5\right)$ of type $\left(8^m:5\right)$. Finally, using Construction 3, we obtain an $S_{\lambda }\left(3,{\mathrm{\Gamma }}_{5,1},8m+5\right)$-design.

$\left(4\right)$ For $v\equiv 5(mod12)$, by decomposition, the set $\{v\mid v\equiv 1(mod12\left)\right\}=\{v\mid v\equiv 1(mod12\left)\right\}\cup \{v\mid v\equiv 5(mod8\left)\right\}$, and $v\ge 17$.

When $v\equiv 1(mod8)$, an $S\left(3,{\mathrm{\Gamma }}_{5,1},v\right)$-design exists. For any positive integer $\lambda$, repeating each block of the $S\left(3,{\mathrm{\Gamma }}_{5,1},v\right)$-design $\lambda$ times yields an $S_{\lambda }\left(3,{\mathrm{\Gamma }}_{5,1},v\right)$-design.

When $v\equiv 5(mod8)$ and $v\ge 17$, let $v=8m+5$, where $m\equiv 0(mod3)$ and $m\ge 3$. By Lemma 4, a $H(m,4,4,3)$-design exists. Using Construction $2.1$, we can obtain a ${\mathrm{GDD}}_2\left(3,{\mathrm{\Gamma }}_{5,1},8m\right)$, and the required ${\mathrm{GDD}}_2\left(3,{\mathrm{\Gamma }}_{5,1},2^4\right)$-design is obtained from Lemma 10. By Lemma 8, an $S_2\left(3,{\mathrm{\Gamma }}_{5,1},21\right)$-design exists, thus a ${\mathrm{CS}}_2\left(3,{\mathrm{\Gamma }}_{5,1},21\right)$-design of type $\left(8^2:5\right)$ certainly exists, and using Construction 2, we can obtain a $CS\left(3,{\mathrm{\Gamma }}_{5,1},8m+5\right)$ of type $\left(8^m:5\right)$. Finally, using Construction 3, we obtain an $S_{\lambda }\left(3,{\mathrm{\Gamma }}_{5,1},8m+5\right)$-design.

$\left(5\right)$ For $v\equiv 9(mod12)$, by decomposition, the set $\{v\mid v\equiv 1(mod12\left)\right\}=\{v\mid v\equiv 1(mod8\left)\right\}\cup \{v\mid v\equiv 5(mod8\left)\right\}$, and $v\ge 21$.

When $v\equiv 1(mod8)$, an $S\left(3,{\mathrm{\Gamma }}_{5,1},v\right)$-design exists. For any positive integer $\lambda$, repeating each block of the $S\left(3,{\mathrm{\Gamma }}_{5,1},v\right)$-design $\lambda$ times yields an $S_{\lambda }\left(3,{\mathrm{\Gamma }}_{5,1},v\right)$-design.

When $v\equiv 5(mod8)$ and $v\ge 17$, let $v=8m+5$, where $m\equiv 2(mod3)$ and $m\ge 2$. By Lemma 5, a 2-$FG(3,(3,3,\{4,6\}),2k)$-design of type $2^{k-2}{4}^1$ exists. Using Construction 4, we can obtain a ${\mathrm{CS}}_2\left(3,{\mathrm{\Gamma }}_{5,1},8m+5\right)$-design of type $\left(8^{k-2}{16}^1:5\right)$, and the required ${\mathrm{CS}}_2\left(3,{\mathrm{\Gamma }}_{5,1},13\right)$-design is obtained from Lemma 3. Using Construction 1, we can obtain a ${\mathrm{GDD}}_2\left(3,{\mathrm{\Gamma }}_{5,1},4^3{8}^1\right)$-design, and the required ${\mathrm{H}}_2(4,2,4,3)$ and ${\mathrm{GDD}}_2\left(3,{\mathrm{\Gamma }}_{5,1},2^3{4}^1\right)$ from Lemma 2 and Lemma 13. Finally, using Construction 3, we obtain an $S_{\lambda }\left(3,{\mathrm{\Gamma }}_{5,1},8m+5\right)$-design. Note the required ${\mathrm{HS}}_2\left(3,{\mathrm{\Gamma }}_{5,1};17,5\right)$ because the ${\mathrm{CS}}_2\left(3,{\mathrm{\Gamma }}_{5,1}17\right)$ and $S_2\left(3,{\mathrm{\Gamma }}_{5,1},5\right)$ of type $\left(8^2:1\right)$ are present.

For any positive integer $\lambda$, repeating each block of the $S_2\left(3,{\mathrm{\Gamma }}_{5,1},v\right)$-design $\lambda /2$ times results in $S_{\lambda }(3,{\mathrm{\Gamma }}_{5,1})$.

This completes the proof. □

Theorem 3. 

If $\lambda \equiv 0\left(\mathit{mod}4\right)$, $v\ge 5$, then there exists an $S_{\lambda }\left(3,{\mathrm{\Gamma }}_{5,1},v\right)$.

Proof of Theorem 3. 

From Theorems 1 and 2, it is known that when $v\equiv 0,2,4($$mod$$6)$ and $v\equiv 1(mod8)$ with $v\ge 6$, an $S_{\lambda }\left(3,{\mathrm{\Gamma }}_{5,1},v\right)$-design exists; when $v\equiv 0,1,2,4,5,6,8,9,10(mod12)$ and $v\ge 5$, an $S_2\left(3,{\mathrm{\Gamma }}_{5,1},v\right)$-design exists, and if we repeat the blocks of $S_2\left(3,{\mathrm{\Gamma }}_{5,1},v\right)$ twice, we obtain an $S_4\left(3,{\mathrm{\Gamma }}_{5,1},v\right)$-design.

When $v\equiv 3(mod4)$, let $v=4m+3,m\ge 1$. By Lemma 5, when $m\equiv 2(mod3)$ and $m\ge 5$, there exists a 2-$FG(3,(3,3,\{4,6\}),2m)$-design of type $2^{m-2}{4}^1$. Using Construction 4, we can obtain a ${\mathrm{CS}}_4\left(3,{\mathrm{\Gamma }}_{5,1},4m+3\right)$-design of type $\left(4^{m-2}{8}^1:3\right)$, and the required ${\mathrm{CS}}_4\left(3,{\mathrm{\Gamma }}_{5,1},7\right)$-design of type $\left(2^3:1\right)$ is obtained from Lemma 3. The required ${\mathrm{GDD}}_4\left(3,{\mathrm{\Gamma }}_{5,1},2^3{4}^1\right)$ is obtained from Construction 1, and finally, using Construction 3, we obtain an $S_4\left(3,{\mathrm{\Gamma }}_{5,1},4m+3\right)$-design. Note that the required ${\mathrm{HS}}_2\left(3,{\mathrm{\Gamma }}_{5,1};7,3\right)$ is because the type $S_4\left(3,{\mathrm{\Gamma }}_{5,1},7\right)$ exists.

When $v\equiv 3(mod4)$, let $v=4m+3,m\ge 1$. By Lemma 5, when $m\equiv 0,1(mod3)$ and $m\ge 3$, there exists a $H(m,4,4,3)$-design. Using Construction 1, we can obtain a $\mathrm{GDD}\left(3,{\mathrm{\Gamma }}_{5,1},4m\right)$-design, and the required ${\mathrm{GDD}}_2\left(3,{\mathrm{\Gamma }}_{5,1},2^4\right)$-design is obtained from Lemma 10. Then, by Lemmas 14 and 15, $S_4\left(3,{\mathrm{\Gamma }}_{5,1},7\right),S_4\left(3,{\mathrm{\Gamma }}_{5,1},11\right)$ exist, and using Construction $2.2$, we can obtain a $CS\left(3,{\mathrm{\Gamma }}_{5,1},4m+3\right)$-design of type $\left(4^m:3\right)$. Finally, using Construction $2.2$, we obtain an $S_{\lambda }\left(3,{\mathrm{\Gamma }}_{5,1},4m+3\right)$-design.

For any positive integer $\lambda$, repeating each block of the $S_4\left(3,{\mathrm{\Gamma }}_{5,1},v\right)$-design $\lambda /4$ times can result in $S_{\lambda }(3,{\mathrm{\Gamma }}_{5,1})$.

This completes the proof. □

Theorem 4. 

The existence of an $S_{\lambda }\left(3,{\mathrm{\Gamma }}_{5,1},v\right)$-design is characterized by the following necessary and sufficient conditions:

$$24\left|\lambda v\right(v-1\left)\right(v-2),2|\lambda (v-1)(v-2),v\ge 5.$$

Proof of Theorem 4. 

Corollary 1 provides the necessary conditions for the existence of an $S_{\lambda }(3,\mathrm{\Gamma },v)$-design. Theorems 1–3 demonstrate the sufficiency of these conditions for the existence of an $S_{\lambda }\left(3,{\mathrm{\Gamma }}_{5,1},v\right)$.

This completes the proof. □

In this paper, we have completely solved the existence problem for decomposing the $\lambda$-fold complete 3-uniform hypergraph, $\lambda K_v^{\left(3\right)}$, into 4-cycles of the specific type ${\mathrm{\Gamma }}_{5,1}$. We established that such a decomposition, denoted as $S_{\lambda }(3,{\mathrm{\Gamma }}_{5,1},v)$-design, exists if and only if $v\ge 5$, $24\left|\lambda v\right(v-1\left)\right(v-2)$, and $2\left|\lambda \right(v-1\left)\right(v-2)$.

This result was achieved through a combination of recursive constructions and direct computational methods for small orders. The recursive frameworks presented (Constructions 1–4) demonstrate how large, symmetric designs can be systematically built from smaller components, leveraging concepts from group-divisible designs and candelabra systems. This highlights a profound interplay between combinatorial design theory and symmetry.

The decomposition of hypergraphs into cycles—particularly into small, symmetric configurations such as 4-cycles—holds fundamental significance in combinatorial design theory and hypergraph theory. Such decompositions not only reveal profound structural symmetries within hypergraphs but also facilitate the construction of designs exhibiting desirable properties, including balance, regularity, and resolvability. As a minimal non-trivial cyclic structure in hypergraphs, the 4-cycle serves as an essential building block for more complex configurations and plays a pivotal role in understanding the combinatorial and algebraic properties of hypergraphs.

The significance of this work is multi-fold. Firstly, it answers a natural and fundamental question in hypergraph decomposition theory, a core area of combinatorics. Secondly, cycles are among the most fundamental sub-structures in graphs and hypergraphs. Understanding the conditions for their existence in decompositions provides crucial insights into the overall architecture of complex hypergraphs. Results like ours often serve as essential building blocks for more complex constructions, aid in solving edge-covering problems, and find applications in areas requiring symmetric partitioning of relational data, such as the design of efficient network codes or distributed storage systems. Finally, just as the analysis of topological indices in chemical graph theory [18] relies on a deep understanding of molecular graph structure, the decomposition of hypergraphs into cycles provides a foundational toolkit for analyzing the properties and symmetries of higher-order networks.