$F^{*}(V(T_{1}))=\mathcal{E}_{1}$ is a _hyperedge set_ defined on $\Lambda$,
and moreover $\\{g(uv)\\}=F^{*}(u)\cap F^{*}(v)\subseteq F^{*}(uv)$.

Step 2. We define a new total set-labelling $F^{*}_{1}$ for $T_{1}$ in the
following steps:

Step 2.1. $F^{*}_{1}(x)=F^{*}(x)$ for each vertex $x\in V(T_{1})$.

Step 2.2. Suppose that $P_{1}=x_{1,1}x_{1,2}\dots x_{1,n_{1}}$ is a longest
path of the tree $T_{1}$ with $m\geq 3$, $x_{1,2}$ and $x_{1,n_{1}-1}$ are not
leaves of $T_{1}$. The neighbor set $N(x_{1,2})$ of the vertex $x_{1,2}$ is of
form as $N(x_{1,2})=\\{x_{1,3}\\}\cup L_{eaf}(x_{1,2})$ for leaf set

$L_{eaf}(x_{1,2})=\\{x_{1,1}\\}\cup\\{y_{1,j}:j\in[1,\textrm{deg}(x_{1,2})-1],y_{1,j}\in
N(x_{1,2})\\}\subset L(T_{1})$

and the neighbor set $N(x_{1,n_{1}-1})=\\{x_{1,n_{1}-2}\\}\cup
L_{eaf}(x_{1,n_{1}-1})$ for leaf set

$L_{eaf}(x_{1,n_{1}-1})=\\{x_{1,n_{1}}\\}\cup\\{u_{1,i}:i\in[1,\textrm{deg}(x_{1,n_{1}-1})-1],u_{1,i}\in
N(x_{1,n_{1}-1})\\}\subset L(T_{1})$

Now, we color these leaf-edges $x_{1,2}y$ and $x_{1,n_{1}-1}u$ as

${\begin{split}&F^{*}_{1}(x_{1,2}y)=F^{*}(x_{1,2}y)\cup
F^{*}(x_{1,2}x_{1,3}),~{}y\in L_{eaf}(x_{1,2})\\\
&F^{*}_{1}(x_{1,n_{1}-1}u)=F^{*}(x_{1,n_{1}-1}u)\cup
F^{*}(x_{1,n_{1}-1}x_{1,n_{1}-2}),~{}u\in L_{eaf}(x_{1,n_{1}-1})\end{split}}$
(25)

Step 2.3. Let $T_{2}=T_{1}-L_{eaf}(x_{1,2})-L_{eaf}(x_{1,n_{1}-1})$ with
$D(T_{2})\geq 3$, and take a longest path $P_{2}=x_{2,1}x_{2,2}\dots
x_{2,n_{2}}$ of the tree $T_{2}$. We have two leaf sets $L_{eaf}(x_{2,2})$ and
$L_{eaf}(x_{2,n_{2}-1})$, and two vertices $x_{2,3},x_{2,n_{2}-2}$ are not in
$L(T_{2})$. We color those leaf-edges $x_{2,2}y$ and $x_{2,n_{2}-1}u$ in the
following

${\begin{split}&F^{*}_{1}(x_{2,2}y)=F^{*}(x_{2,2}y)\cup
F^{*}(x_{2,2}x_{2,3}),~{}y\in L_{eaf}(x_{2,2})\\\
&F^{*}_{1}(x_{2,n_{2}-1}u)=F^{*}(x_{2,n_{2}-1}u)\cup
F^{*}(x_{2,n_{2}-1}x_{2,n_{2}-2}),~{}u\in L_{eaf}(x_{2,n_{2}-1})\end{split}}$
(26)

Step 2.$k+2$. We have a tree
$T_{k+1}=T_{k}-L_{eaf}(x_{k,2})-L_{eaf}(x_{k,n_{k}-1})$ with $D(T_{k+1})\geq
3$ and $k\in[1,m-1]$, and take a longest path $P_{k+1}=x_{k+1,1}x_{k+1,2}\dots
x_{k+1,n_{k+1}}$ of the tree $T_{k+1}$. There are two leaf sets
$L_{eaf}(x_{k+1,2})$ and $L_{eaf}(x_{k+1,n_{k+1}-1})$, and two vertices
$x_{k+1,3},x_{k+1,n_{k+1}-2}$ are not in $L(T_{k+1})$. We color those leaf-
edges $x_{k+1,2}y$ and $x_{k+1,n_{k+1}-1}u$ as follows

${\begin{split}&F^{*}_{1}(x_{k+1,2}y)=F^{*}(x_{k+1,2}y)\cup
F^{*}(x_{k+1,2}x_{k+1,3}),y\in L_{eaf}(x_{k+1,2})\\\
&F^{*}_{1}(x_{k+1,n_{k+1}-1}u)=F^{*}(x_{k+1,n_{k+1}-1}u)\cup
F^{*}(x_{k+1,n_{k+1}-1}x_{k+1,n_{k+1}-2}),u\in
L_{eaf}(x_{k+1,n_{k+1}-1})\end{split}}$ (27)

Thereby, we have recolored the edges of the tree $T_{1}$ well, and it is not
hard to see $F^{*}_{1}(uv)\cap F^{*}_{1}(uw)\neq\emptyset$ for $w\in N(u)$ and
each vertex $u\in V(T_{1})$, see examples shown in Fig.23. We claim that
$F^{*}_{1}$ is an adjacent edge-intersected total set-labeling of $T_{1}$.

The proof of the theorem is complete. ∎

Figure 23: Another algorithm for obtaining the adjacent edge-intersected total
set-labeling $F^{*}$ of the tree $O_{1}$ shown in Fig.22 (a), where
$F^{*}(uv)=\\{8\\}$ in the set-colored tree $R_{5}$, and there are two perfect
hypermatchings in the intersected-graph $R_{1}$.

######  Theorem 32.

Each connected graph admits one of subintersected total set-labeling,
intersected-edge-intersected total set-labeling, edge-intersected total set-
labeling, adjacent edge-intersected total set-labeling and individual edge-
intersected total set-labeling defined in Definition 24.

###### Proof.

By the vertex-splitting operation, we vertex-split a connected $(p,q)$-graph
$G$ into a tree $T_{G}$ of $(q+1)$ vertices. Since $T_{G}$ admits one of
subintersected total set-labeling, intersected-edge-intersected total set-
labeling, edge-intersected total set-labeling, adjacent edge-intersected total
set-labeling and individual edge-intersected total set-labeling according to
Theorem 31.

Doing the vertex-coinciding operation to $T_{G}$ produces the original
connected $(p,q)$-graph $G$, we define a set-coloring $F^{*}$ for $G$ in the
following: Since $T_{G}$ admits a set-coloring
$F:V(T_{G})\rightarrow\mathcal{E}$ to be one intersected-type total set-
labeling defined in Definition 24, we define the desired set-coloring $F^{*}$
as: $F^{*}(z)=F(x)\cup F(y)$ if $z=x\odot y$ for $z\in V(G)$ and $x,y\in
V(T_{G})$, $F^{*}(uv)=F(uv)$ for $uv\in E(G)$ and $uv\in E(T_{G})$, then we
claim that $G$ admits each one of the intersected-type total set-labelings
defined in Definition 24.

The proof of the theorem is completed. ∎

##  5 Graph operations on intersected-graphs and hypergraphs

###  5.1 Set-increasing operation and set-decreasing operations

Let $\mathcal{E}=\\{e_{1}$, $e_{2}$, $\dots$, $e_{n}\\}=\\{e_{i}\\}^{n}_{i=1}$
and
$\mathcal{X}=\\{e^{*}_{1},e^{*}_{2},\dots,e^{*}_{n}\\}=\\{e^{*}_{i}\\}^{n}_{i=1}$
be two hyperedge sets of subsets of a finite set $\Lambda$. We do a set
subtraction operation to $\mathcal{E}$ and $\mathcal{X}$, such that each set
$e\,^{\prime}_{i}$ of the resultant subset set
$\mathcal{E}\,^{\prime}=\\{e\,^{\prime}_{i}\\}^{n}_{i=1}$ holds
$e\,^{\prime}_{i}=e_{i}\setminus e^{*}_{i_{j}}$ or $e\,^{\prime}_{i}=e_{i}$
with $e_{i}\in\mathcal{E}$ and $e^{*}_{i_{j}}\in\mathcal{X}$ and $i\in[1,n]$,
and there is at least a set $e\,^{\prime}_{j}$ holding $e\,^{\prime}_{j}\neq
e_{j}$ for some $j$. We write
$\mathcal{E}\,^{\prime}=\mathcal{E}[\setminus]\mathcal{X}$ and
$\mathcal{E}=\mathcal{E}\,^{\prime}[\cup]\mathcal{X}$, respectively. And we
call $\mathcal{H}\,^{\prime}_{yper}=(\Lambda,\mathcal{E}\,^{\prime})$ _set-
decreased hypergraph_ of the hypergraph
$\mathcal{H}_{yper}=(\Lambda,\mathcal{E})$ according to
$\mathcal{E}\,^{\prime}=\mathcal{E}[\setminus]\mathcal{X}$; conversely,
$\mathcal{H}_{yper}$ is a _set-increased hypergraph_ of the hypergraph
$\mathcal{H}\,^{\prime}_{yper}$ because of
$\mathcal{E}=\mathcal{E}\,^{\prime}[\cup]\mathcal{X}$.

######  Definition 25.

∗ For the hyperedge set $\mathcal{E}$ of a hypergraph
$\mathcal{H}_{yper}=(\Lambda,\mathcal{E})$ if there is no hyperedge set
$\mathcal{X}\subset\Lambda^{2}$ such that $\mathcal{E}[\setminus]\mathcal{X}$
is the hyperedge set $\mathcal{E}^{*}$ of some hypergraph
$\mathcal{H}^{*}_{yper}=(\Lambda,\mathcal{E}^{*})$, then we say that the
hyperedge set $\mathcal{E}$ defined on $\Lambda$ is not _decreasing_.
$\square$

######  Example 9.

A hyperedge set $\mathcal{E}$ defined in Eq.(13) and another hyperedge set
$\mathcal{X}=\\{\\{1\\}$, $\\{2\\}$, $\\{7\\}$, $\\{8\\}\\}$ produce the
following hyperedge set

${\begin{split}\mathcal{E}\,^{\prime}=\mathcal{E}[\setminus]\mathcal{X}=&\big{\\{}\\{12\\},\\{11\\},\\{10\\},\\{6,10,11,12\\},\\{4,5,6\\},\\{5,7\\},\\\
&\\{7,8,9\\},\\{8\\},\\{4,9\\},\\{3,4\\},\\{2\\},\\{1,2,3\\},\\{1\\}\big{\\}}\end{split}}$
(28)

and $\bigcup_{e_{i}\in\mathcal{E}\,^{\prime}}e_{i}=[1,12]$, so we get a
hypergraph $\mathcal{H}\,^{\prime}_{yper}=([1,12],\mathcal{E}\,^{\prime})$
defined by the hyperedge set $\mathcal{E}\,^{\prime}$ shown in Eq.(28) and its
intersected-graph $L$ shown in Fig.24 (a). The hypergraph
$\mathcal{H}\,^{\prime}_{yper}=([1,12],\mathcal{E}\,^{\prime})$ has a _perfect
hypermatching_
$\big{\\{}\\{12\\},\\{11\\},\\{10\\},\\{4,5,6\\},\\{7,8,9\\},\\{1,2,3\\}\big{\\}}$,
which is one of the intersected-graph $L$ too. Clearly, the Graham reduction
of the hyperedge set $\mathcal{E}\,^{\prime}$ is an empty set, so
$\mathcal{H}\,^{\prime}_{yper}$ is acyclic, on the other hand, the
intersected-graph $L$ is a tree, which implies that
$\mathcal{H}\,^{\prime}_{yper}$ is acyclic.

Since the intersected-graph $L$ is a tree, so the corresponding hypergraph
$\mathcal{H}\,^{\prime}_{yper}$ is _acyclic_ , and $L$ is the unique
intersected-graph of the hypergraph $\mathcal{H}\,^{\prime}_{yper}$. Moreover,
$\mathcal{H}\,^{\prime}_{yper}$ is a set-decreased hypergraph of the
hypergraph $\mathcal{H}_{yper}$ defined by the hyperedge set $\mathcal{E}$
shown in Eq.(13), and the intersected-graph $L$ is a subgraph of the
intersected-graph $G_{yper}$ shown in Fig.9 (b). $\square$

Notice that the hyperedge set $\mathcal{E}\,^{\prime}$ defined in Eq.(28) is
not decreasing, we have the following result:

######  Theorem 33.

If the hyperedge set $\mathcal{E}$ of a hypergraph
$\mathcal{H}_{yper}=(\Lambda,\mathcal{E})$ is not decreasing, then this
hypergraph $\mathcal{H}_{yper}$ is acyclic, also, its intersected-graph is
acyclic too.

######  Remark 4.

A set-colored graph $G$ may be not the intersected-graph of any hypergraph,
however, some proper subgraph of $G$ is really the intersected-graph of some
hypergraph, see a set-colored graph $L_{2}$ shown in Fig.24. $\parallel$

Figure 24: (a) A tree $L$ admitting a graceful-intersection total set-labeling
defined on the hyperedge set $\mathcal{E}\,^{\prime}$ shown in Eq.(28) is an
intersected-graph; (b) and (c) are the resultant graphs of doing the vertex-
splitting operation to $L$, in which $L_{2}$ is an intersected-graph based on
a hyperedge set $\big{\\{}\\{1,2,3\\}$, $\\{1\\}$, $\\{2\\}$, $\\{3,4\\}$,
$\\{4,5,6\\}$, $\\{5,7\\}$, $\\{7,8,9\\}$, $\\{8\\}$,
$\\{9\\}\big{\\}}\subset[1,9]^{2}$.

###  5.2 Splitting-type and coinciding-type operations

Suppose that the intersected-graph $H$ of a hypergraph
$\mathcal{H}_{yper}=(\Lambda,\mathcal{E})$ subject to $R_{est}(m+1)$ admits a
set-coloring $F:V(H)\rightarrow\mathcal{E}$, such that each edge $uv$ of
$E(H)$ is colored with an induced edge color $F(uv)$ defined in Definition 20.
We introduce the following basic splitting and coinciding operations on the
vertices and edges of intersected-graphs or graphs.

####  5.2.1 Edge-splitting and edge-coinciding operations of intersected-
graphs

Let $G$ be an intersected-$(p,q)$-graph admitting a $W$-type set-coloring $F$
defined on a hypergraph $\mathcal{H}_{yper}=(\Lambda,\mathcal{E})$. We edge-
split an edge $uv\in E(G)$ into two edges $u\,^{\prime}v\,^{\prime}$ and
$u\,^{\prime\prime}v\,^{\prime\prime}$, such that $N(u)=N(u\,^{\prime})\cup
N(u\,^{\prime\prime})$ and $N(v)=N(v\,^{\prime})\cup N(v\,^{\prime\prime})$,
the resultant graph is denoted as $G\wedge uv$, which has $|V(G\wedge
uv)|=|V(G)|+2=p+2$ vertices and $|E(G\wedge uv)|=|E(G)|+1=q+1$ edges. We
define a set-coloring $F^{*}$ of the edge-split graph $G\wedge uv$ as:

  1. Esc-1

$F^{*}(z)=F(z)$ for $z\in V(G-\\{u,uv,v\\})\cup E(G-\\{u,uv,v\\})$.

  2. Esc-2

$F^{*}(xu\,^{\prime})=F(xu)$ for $x\in N(u\,^{\prime})\subset N(u)$,
$F^{*}(yu\,^{\prime\prime})=F(yu)$ for $y\in N(u\,^{\prime\prime})\subset
N(u)$.

  3. Esc-3

$F^{*}(wv\,^{\prime})=F(wv)$ for $w\in N(v\,^{\prime})\subset N(v)$,
$F^{*}(zv\,^{\prime\prime})=F(zv)$ for $z\in N(v\,^{\prime\prime})\subset
N(v)$.

  4. Esc-4

$F^{*}(u\,^{\prime})=F(u)$ and $F^{*}(u\,^{\prime\prime})=F(u)$,
$F^{*}(v\,^{\prime})=F(v)$ and $F^{*}(v\,^{\prime\prime})=F(v)$.

  5. Esc-5

$F^{*}(u\,^{\prime}v\,^{\prime})=F(uv)$ and
$F^{*}(u\,^{\prime\prime}v\,^{\prime\prime})=F(uv)$.

Thereby, the edge-split $(p+2,q+1)$-graph $G\wedge uv$ admitting a $W$-type
set-coloring $F^{*}$ is the intersected-$(p+2,q+1)$-graph of a hypergraph
$\mathcal{H}_{yper}=(\Lambda^{*},\mathcal{E}^{*})$, where
$\Lambda^{*}=\Lambda$ and $\mathcal{E}^{*}=\mathcal{E}$.

Conversely, we have the original intersected-graph
$G=H[u\,^{\prime}v\,^{\prime}\ominus u\,^{\prime\prime}v\,^{\prime\prime}]$ by
the edge-coinciding operation, where $H=G\wedge uv$.

######  Problem 14.

If two hypergraphs $\mathcal{H}_{yper}=(\Lambda,\mathcal{E})$ and
$\mathcal{H}_{yper}=(\Lambda^{*},\mathcal{E}^{*})$ hold $\Lambda^{*}=\Lambda$
and $\mathcal{E}^{*}=\mathcal{E}$, characterize their intersected-graphs.

Suppose that $H_{1}$ and $H_{2}$ are two vertex-disjoint graphs and each
$H_{i}$ is the intersected-graph of the hypergraph
$\mathcal{H}^{i}_{yper}=(\Lambda_{i},\mathcal{E}_{i})$ for $i=1,2$. We take
edges $u_{i,j}v_{i,j}\in E(H_{i})$ for $i=1,2$ and $j\in[1,s]$, and do the
edge-coinciding operation to edges $u_{1,j}v_{1,j}$ and $u_{2,j}v_{2,j}$ in
order to obtain edge-coincided edges $u_{j}v_{j}=u_{1,j}v_{1,j}\ominus
u_{2,j}v_{2,j}$ with vertex-coincided vertices $u_{j}=u_{1,j}\odot u_{2,j}$,
$v_{j}=v_{1,j}\odot v_{2,j}$ for $j\in[1,s]$, the resultant graph is denoted
as $\ominus\langle H_{1},H_{2}\rangle$.

Suppose that each intersected-graph $H_{i}$ admits a set-coloring $F_{i}$
defined on the hyperedge set $\mathcal{E}_{i}$ based on a finite set
$\Lambda_{i}$ with $i=1,2$, and then we define a set-coloring $F$ for
$\ominus\langle H_{1},H_{2}\rangle$ by setting $F(u_{j})=F_{1}(u_{1,j})\cup
F_{2}(u_{2,j})$, $F(v_{j})=F_{1}(v_{1,j})\cup F_{2}(v_{2,j})$ and
$F(u_{j}v_{j})=F_{1}(u_{1,j}v_{1,j})\cup F_{2}(u_{2,j}v_{2,j})$ with
$j\in[1,s]$, and

$F(w)=F_{i}(w),~{}w\in\big{[}V(H_{i})\cup
E(H_{i})\big{]}\setminus\big{[}\\{u_{i,j},v_{i,j}:j\in[1,s]\\}\cup\\{u_{i,j}v_{i,j}:j\in[1,s]\\}\big{]}$

with $i=1,2$. So, $\ominus\langle H_{1},H_{2}\rangle$ is the intersected-graph
of a hypergraph
$\mathcal{H}^{\ominus}_{yper}=\big{(}\Lambda_{1}\cup\Lambda_{2},\mathcal{E}_{1}\cup\mathcal{E}_{2}\big{)}$,
where

$\mathcal{H}^{\ominus}_{yper}=\ominus\langle\mathcal{H}^{1}_{yper},\mathcal{H}^{2}_{yper}\rangle=\big{(}\Lambda_{1}\cup\Lambda_{2},\mathcal{E}_{1}\cup\mathcal{E}_{2}\big{)}$
(29)

In general, let $\textbf{{H}}=(H_{1},H_{2},\dots,H_{m})$ be an _intersected-
graph base_ , where each intersected-graph $H_{i}$ admits a set-coloring
$F_{i}$ defined on the hyperedge set $\mathcal{E}_{i}$ of the hypergraph
$\mathcal{H}^{i}_{yper}=(\Lambda_{i},\mathcal{E}_{i})$ with $i\in[1,m]$. Let
$G_{1},G_{2},\dots,G_{A}$ be a permutation of intersected-graphs
$a_{1}H_{1},a_{2}H_{2},\dots,a_{m}H_{m}$, where $A=\sum^{m}_{k=1}a_{k}\geq 1$.
Now, we do the edge-coinciding operation to edges $x_{k,j}y_{k,j}\in E(G_{k})$
and $x_{k+1,j}y_{k+1,j}\in E(G_{k+1})$ for $k\in[1,A-1]$ to obtain a graph

$L=\ominus\langle G_{1},G_{2},\dots,G_{A}\rangle=\ominus^{m}_{k=1}\langle
a_{k}H^{k}\rangle$ (30)

to be a subgraph of the intersected-graph of a hypergraph

$\mathcal{H}^{\ominus}_{yper}(L)=\big{(}\Lambda(L),\mathcal{E}(L)\big{)}=\ominus^{m}_{k=1}\langle
a_{k}\mathcal{H}^{k}_{yper}\rangle$ (31)

where $\Lambda(L)=\bigcup^{m}_{k=1}\Lambda_{k}$ and
$\mathcal{E}(L)=\bigcup^{m}_{k=1}\mathcal{E}_{k}$.

We get an _edge-coincided intersected-graph lattice_ as

$\textbf{{L}}\big{(}Z^{0}\ominus\textbf{{H}}\big{)}=\left\\{\ominus^{m}_{k=1}\langle
a_{k}H_{k}\rangle:a_{k}\in Z^{0},H_{k}\in\textbf{{H}}\right\\}$ (32)

with $\sum^{m}_{k=1}a_{k}\geq 1$. Clearly, each set-colored graph
$L\in\textbf{{L}}\big{(}Z^{0}\ominus\textbf{{H}}\big{)}$ can be decomposed
into vertex disjoint intersected-graphs
$a_{1}H_{1},a_{2}H_{2},\dots,a_{m}H_{m}$ by the vertex-splitting operation.

By Eq.(31) and Eq.(32), we have a _hyperedge-coincided hypergraph lattice_ as
follows

$\textbf{{L}}\big{(}Z^{0}\ominus\textbf{{H}}_{yper}\big{)}=\left\\{\ominus^{m}_{k=1}\langle
a_{k}\mathcal{H}^{k}_{yper}\rangle:a_{k}\in
Z^{0},\mathcal{H}^{k}_{yper}\in\textbf{{H}}_{yper}\right\\}$ (33)

where
$\textbf{{H}}_{yper}=(\mathcal{H}^{1}_{yper},\mathcal{H}^{2}_{yper},\dots,\mathcal{H}^{m}_{yper})$
is the _hypergraph lattice base_.

######  Theorem 34.

In the intersected-graph base H of
$\textbf{{L}}\big{(}Z^{0}\ominus\textbf{{H}}\big{)}$ defined in Eq.(32), if
each intersected-graph $H_{i}$ is a tree, then $L=\ominus^{m}_{k=1}\langle
a_{k}H_{k}\rangle$ is a tree too, and moreover if $\Lambda=\Lambda_{i}$ for
$i\in[1,m]$, $\Lambda(L)=\Lambda$, then $L$ is the intersected-graph of a
hypergraph
$\mathcal{H}^{\ominus}_{yper}=\big{(}\Lambda,\bigcup^{m}_{k=1}\mathcal{E}_{k}\big{)}$.

####  5.2.2 Vertex-splitting and vertex-coinciding operations of intersected-
graphs

We introduce the concept of vertex-split graph in the following way: Select
randomly a subset $X\subset V(H)$, where the graph $H$ admits a set-coloring
$F:V(H)\rightarrow\mathcal{E}$ defined on a hypergraph
$\mathcal{H}_{yper}=(\Lambda,\mathcal{E})$, and do the vertex-splitting
operation to each vertex $w_{i}\in X$, such that $w_{i}$ is split into
$w\,^{\prime}_{i}$ and $w\,^{\prime\prime}_{i}$, and
$N(w_{i})=N(w\,^{\prime}_{i})\cup N(w\,^{\prime\prime}_{i})$ with
$N(w\,^{\prime}_{i})\cap N(w\,^{\prime\prime}_{i})=\emptyset$. The resultant
graph is denoted as $H\wedge X$, called _vertex-split graph_. Now, we define a
set-coloring $F^{*}$ by $F$ for the vertex-split graph $H\wedge X$ as follows:

  1. Vsc-1.

$F^{*}(z)=F(z)$ for $z\in V(H-X)\cup E(H-X)$.

  2. Vsc-2.

$F^{*}(xw\,^{\prime}_{i})=F(xw\,^{\prime}_{i})$ for $x\in
N(w\,^{\prime}_{i})$.

  3. Vsc-3.

$F^{*}(xw\,^{\prime\prime}_{i})=F(xw\,^{\prime\prime}_{i})$ for $x\in
N(w\,^{\prime\prime}_{i})$.

  4. Vsc-4.

$F^{*}(w\,^{\prime}_{i})=F(w_{i})$ and
$F^{*}(w\,^{\prime\prime}_{i})=F(w_{i})$.

  5. Vsc-5.

$F(w_{i})=F^{*}(w\,^{\prime}_{i})\cup F^{*}(w\,^{\prime\prime}_{i})$ and
$F^{*}(w\,^{\prime}_{i})\cap F(w\,^{\prime\prime}_{i})=\emptyset$.

  6. Vsc-6.

$F^{*}(w\,^{\prime}_{i})\subset F(w_{i})$ and
$F^{*}(w\,^{\prime\prime}_{i})\subset F(w_{i})$, and
$F^{*}(w\,^{\prime}_{i})\cap F^{*}(w\,^{\prime\prime}_{i})\neq\emptyset$.

In Fig.24, we have a vertex-split graph $L\wedge u$, which has two components
$L_{1}$ and $L_{2}$ shown in Fig.24 (b) and (c), we call $L_{1}$ and $L_{2}$
two _partial hypergraphs_ of the hypergraph having its intersected-graph
$L=\odot\langle L_{1},L_{2}\rangle$.

If the vertex-split graph $H\wedge X$ is disconnected, that is, $H\wedge X$
consists of vertex-disjoint components $H_{1},H_{2},\dots,H_{s}$ with $s\geq
2$, and $H\wedge X$ admits a set-coloring $F^{*}$ holding Vsc-1, Vsc-2, Vsc-3
and Vsc-4, we call each $H_{i}$ a _partial hypergraph_ of the hypergraph

$\mathcal{H}_{yper}=(\Lambda,\mathcal{E})=\left(\bigcup^{m}_{i=1}\Lambda_{i},\bigcup^{s}_{i=1}\mathcal{E}_{i}\right)$
(34)

where each hypergraph $\mathcal{H}^{i}_{yper}=(\Lambda_{i},\mathcal{E}_{i})$
for $i\in[1,s]$.

We do the vertex-coinciding operation to the vertex-split graph $H\wedge X$
admitting the set-coloring $F^{*}$ by vertex-coinciding each pair of vertices
$w\,^{\prime}_{i}$ and $w\,^{\prime\prime}_{i}$ into one vertex
$w_{i}=w\,^{\prime}_{i}\odot w\,^{\prime\prime}_{i}$, and make
$F(w_{i})=F^{*}(w\,^{\prime}_{i})\cup F^{*}(w\,^{\prime\prime}_{i})$, such
that the resultant graph is just the original graph $H$, we write
$H=\odot\langle H_{1},H_{2}$, $\dots$, $H_{s}\rangle=\odot\langle
H_{i}\rangle^{s}_{i=1}$, or $H=\odot[H\wedge X]$.

DC-algorithm for the decomposition and composition of hypergraphs. We do the
vertex-coinciding operation to vertex-disjoint intersected-graphs
$G_{1},G_{2},\dots,G_{n}$ for getting a new graph

$G^{*}=\odot\langle G_{1},G_{2},\dots,G_{n}\rangle=\odot\langle
G_{k}\rangle^{n}_{k=1}$

Suppose that each intersected-graph $G_{j}$ admits a set-coloring $F_{j}$
defined on a hyperedge set $\mathcal{E}_{j}$, we define a set-coloring $F^{*}$
of $G^{*}$ as:

  1. Op-1

If $z_{i,j}=z_{i}\odot z_{j}$ for $z_{i}\in V(G_{i})$ and $z_{j}\in V(G_{j})$
with $i\neq j$, set $F^{*}(z_{i,j})=F_{i}(z_{i})\cup F_{j}(z_{j})$.

  2. Op-2

If $z_{i,j}=z_{i}\odot z_{j}$ and $w_{i,j}=w_{i}\odot w_{j}$ for
$z_{i},w_{i}\in V(G_{i})$ and $z_{j},w_{j}\in V(G_{j})$ with $i\neq j$, and
$z_{i}w_{i}$ is an edge of $G_{i}$ and $z_{j}w_{j}$ is an edge of $G_{j}$, we
obtain a coincided edge $z_{i,j}w_{i,j}=z_{i}w_{i}\odot z_{j}w_{j}$ by doing
the edge-coinciding operation on these two edges $z_{i}w_{i}$ and
$z_{j}w_{j}$, and moreover we set

$F^{*}(z_{i,j})=F_{i}(z_{i})\cup F_{j}(z_{j}),F^{*}(w_{i,j})=F_{i}(w_{i})\cup
F_{j}(w_{j}),F^{*}(z_{i,j}w_{i,j})=F_{i}(z_{i}w_{i})\cup F_{j}(z_{j}w_{j})$

Since each intersected-graph $G_{j}$ holds $F_{j}(V(G_{j}))=\mathcal{E}_{j}$
in the hypergraph $\mathcal{H}^{j}_{yper}=(\Lambda_{j},\mathcal{E}_{j})$, then
$F^{*}(V(G^{*}))=\mathcal{E}^{*}=\bigcup^{n}_{j=1}\mathcal{E}_{j}$. We have
the following particular hypergraphs generated from the graph $G^{*}$:

  1. Hyper-1

If $\Lambda=\Lambda_{j}$ for $j\in[1,n]$, then $\mathcal{E}^{*}$ is a
hyperedge set based on $\Lambda$, immediately, we get a hypergraph
$\mathcal{H}_{yper}=(\Lambda,\mathcal{E}^{*})$.

  2. Hyper-2

If $\Lambda^{*}=\bigcup_{e\in\mathcal{E}^{*}}e$, then $G^{*}$ is the
intersected-graph of the hypergraph
$\mathcal{H}_{yper}=(\Lambda^{*},\mathcal{E}^{*})$.

We have an example $\odot\langle
T,T_{1},T_{2},T_{3}\rangle\subseteq\mathcal{H}_{yper}$, where the graphs
$T,T_{1},T_{2},T_{3}$ and the intersected-graph $\mathcal{H}_{yper}$ shown in
Fig.9 and Fig.10.

Figure 25: An example for the _hyperedgeset-splitting_ and _hyperedgeset-
coinciding_ operations.

We consider to build up _vertex-coincided intersected-graph lattices_ in the
following: Let $\textbf{{T}}=(T_{1},T_{2}$, $\dots$, $T_{n})$ be an
_intersected-graph base_ , where each intersected-graph $T_{j}$ admits a set-
coloring $\varphi_{j}$ defined on the hyperedge $\mathcal{E}_{j}$ of each
hypergraph $\mathcal{T}^{j}_{yper}=(\Lambda_{j},\mathcal{E}_{j})$ with
$j\in[1,n]$, and let $J_{1},J_{2},\dots,J_{B}$ be a permutation of
intersected-graphs $b_{1}T_{1},b_{2}T_{2}$, $\dots$, $b_{n}T_{n}$, where
$B=\sum^{n}_{k=1}b_{k}\geq 1$. Now, we do the vertex-coinciding operation to
graphs $J_{1},J_{2},\dots,J_{B}$, the resultant graph

$I=\odot\langle J_{1},J_{2},\dots,J_{B}\rangle=\odot^{n}_{k=1}\langle
b_{k}T_{k}\rangle$ (35)

is a subgraph of the intersected-graph of a hypergraph
$\mathcal{H}^{\odot}_{yper}(I)=(\Lambda(I),\mathcal{E}(I))$, where
$\Lambda(I)=\bigcup^{m}_{k=1}\Lambda_{k}$ and
$\mathcal{E}(I)=\bigcup^{m}_{k=1}\mathcal{E}_{k}$. We get a _vertex-coincided
intersected-graph lattice_

$\textbf{{L}}\big{(}Z^{0}\odot\textbf{{T}}\big{)}=\left\\{\odot^{n}_{k=1}\langle
b_{k}T_{k}\rangle:b_{k}\in Z^{0},T_{k}\in\textbf{{T}}\right\\}$ (36)

with $\sum^{n}_{k=1}b_{k}\geq 1$, such that each set-colored graph
$G\in\textbf{{L}}\big{(}Z^{0}\odot\textbf{{T}}\big{)}$ can be decomposed into
edge-disjoint intersected-graphs $b_{1}T_{1},b_{2}T_{2}$, $\dots$,
$b_{n}T_{n}$, however, it is not easy to realize the decomposition of a graph
to some particular graphs, in general.

Correspondingly, by Eq.(35) and Eq.(36), we have a _hyperedge-coincided
hypergraph lattice_

$\textbf{{L}}\big{(}Z^{0}\odot\textbf{{T}}_{yper}\big{)}=\left\\{\odot^{n}_{k=1}\langle
b_{k}\mathcal{T}^{k}_{yper}\rangle:a_{k}\in
Z^{0},\mathcal{T}^{k}_{yper}\in\textbf{{T}}_{yper}\right\\}$ (37)

with $\sum^{n}_{k=1}b_{k}\geq 1$, where
$\textbf{{T}}_{yper}=(\mathcal{T}^{1}_{yper},\mathcal{T}^{2}_{yper},\dots,\mathcal{T}^{n}_{yper})$
is a _hypergraph lattice base_.

Moreover, we do mixed operations of the vertex-coinciding operation and the
edge-coinciding operation to a permutation $I_{1},I_{2},\dots,I_{B}$ of
intersected-graphs $c_{1}T_{1},c_{2}T_{2}$, $\dots$, $c_{n}T_{n}$, the
resultant graph is written as

$[\odot\ominus]\langle
I_{1},I_{2},\dots,I_{B}\rangle=[\odot\ominus]^{n}_{k=1}\langle
c_{k}T_{k}\rangle$ (38)

with $\sum^{n}_{k=1}c_{k}\geq 1$. We have a _mixed intersected-graph lattice_

$\textbf{{L}}\big{(}Z^{0}[\odot\ominus]\textbf{{T}}\big{)}=\left\\{[\odot\ominus]^{n}_{k=1}\langle
c_{k}T_{k}\rangle:c_{k}\in Z^{0},T_{k}\in\textbf{{T}}\right\\}$ (39)

and a _mixed-coincided hypergraph lattice_

$\textbf{{L}}\big{(}Z^{0}[\odot\ominus]\textbf{{T}}_{yper}\big{)}=\left\\{[\odot\ominus]^{n}_{k=1}\langle
b_{k}\mathcal{T}^{k}_{yper}\rangle:a_{k}\in
Z^{0},\mathcal{T}^{k}_{yper}\in\textbf{{T}}_{yper}\right\\}$ (40)

with $\sum^{n}_{k=1}c_{k}\geq 1$ and the hypergraph lattice base
$\textbf{{T}}_{yper}=(\mathcal{T}^{1}_{yper},\mathcal{T}^{2}_{yper},\dots,\mathcal{T}^{n}_{yper})$.

######  Theorem 35.

In a hyperedge-coincided hypergraph lattice
$\textbf{{L}}\big{(}Z^{0}\odot\textbf{{T}}_{yper}\big{)}$ defined in Eq.(37),
if $\Lambda=\Lambda_{1}=\Lambda_{2}=\cdots=\Lambda_{m}$, and any pair of two
sets $e_{i}\in\mathcal{E}_{i}$ and $e_{j}\in\mathcal{E}_{j}$ hold $e_{i}\cap
e_{j}=\emptyset$ as $i\neq j$, each intersected-graph $H_{i}$ is connected.
Then each graph $G=\ominus^{m}_{k=1}\langle a_{k}H_{k}\rangle$ with $a_{k}=1$
or $0$ defined in Eq.(32) is the intersected-graph of the hypergraph
$\mathcal{H}_{yper}=\big{(}\Lambda,\bigcup^{m}_{i=1}\mathcal{E}_{i}\big{)}$,
so is the graph $I=\odot^{n}_{k=1}\langle b_{k}T_{k}\rangle$ with $b_{k}=1$ or
$0$ defined in Eq.(36), and so is the graph $[\odot\ominus]^{n}_{k=1}\langle
c_{k}T_{k}\rangle$ with $c_{k}=1$ or $0$ defined in Eq.(39).

######  Remark 5.

The number of intersected-graphs $G=\ominus^{m}_{k=1}\langle
a_{k}H_{k}\rangle$ with $a_{k}=1$ or $0$ is $2^{m}$ in total, so are the
numbers of intersected-graphs $I=\odot^{n}_{k=1}\langle b_{k}T_{k}\rangle$
with $b_{k}=1$ or $0$ and $[\odot\ominus]^{n}_{k=1}c_{k}T_{k}$ with $c_{k}=1$
or $0$ in Theorem 35.

The above operations lead to the _hyperedgeset-splitting_ operation and the
_hyperedgeset-coinciding_ operation of hypergraphs. It is noticeable, there
are many operations on hypergraphs for building up various hypergraph
lattices. Planting some problems of traditional lattices into hypergraph
lattices will be important and meaningful in asymmetric cryptography.
$\parallel$

######  Theorem 36.

∗ Suppose that two connected graphs $G$ and $H$ admit a mapping
$f:V(G)\rightarrow V(H)$, where $G$ is the intersected-graph of a hypergraph
$\mathcal{H}_{yper}=(\Lambda,\mathcal{E})$, and $H$ is the intersected-graph
of another hypergraph $\mathcal{H}^{*}_{yper}=(\Lambda^{*},\mathcal{E}^{*})$.
Vertex-splitting a vertex $u$ of the intersected-graph $G$ with
$\textrm{deg}_{G}(u)\geq 2$ produces a vertex-split graph set
$I_{G}(u)=\\{G\wedge u\\}$ (resp. a hypergraph set
$\\{\mathcal{H}_{yper}\wedge e\\}$ with $e=F(u)$, since
$F:V(G)\rightarrow\mathcal{E}$). Similarly, another vertex-split graph set
$I_{H}(f(u))=\\{H\wedge f(u)\\}$ is obtained by vertex-splitting a vertex
$f(u)$ of the intersected-graph $H$ with $\textrm{deg}_{H}(f(u))\geq 2$, so we
have a hypergraph set $\\{\mathcal{H}^{*}_{yper}\wedge e\,^{\prime}\\}$ with
$e\,^{\prime}=F\,^{\prime}(f(u))$, since
$F\,^{\prime}:V(H)\rightarrow\mathcal{E}^{*}$. If each vertex-split graph
$L\in I_{G}(u)$ (resp. $\mathcal{L}\in\\{\mathcal{H}_{yper}\wedge e\\}$)
corresponds another vertex-split graph $T\in I_{H}(f(u))$ (resp.
$\mathcal{T}\in\\{\mathcal{H}^{*}_{yper}\wedge e\,^{\prime}\\}$) such that
$L\cong T$ (resp. $\mathcal{L}\cong\mathcal{T}$), and vice versa, we write
this fact as

$G\wedge u\cong H\wedge f(u)~{}(\textrm{resp.}~{}\mathcal{H}_{yper}\wedge
e\cong\mathcal{H}^{*}_{yper}\wedge e\,^{\prime})$ (41)

then $G\cong H$ (resp. $\mathcal{H}_{yper}\cong\mathcal{H}^{*}_{yper}$).

Recall Kelly-Ulam’s Reconstruction Conjecture (1942): “ _Let both $G$ and $H$
be graphs with $n$ vertices. If there is a bijection $f:V(G)\rightarrow V(H)$
such that $G-u\cong H-f(u)$ for each vertex $u\in V(G)$, then $G\cong H$_.” We
present the following conjectures:

######  Conjecture 1.

∗ Let both $G$ and $H$ be graphs with $n$ vertices. If each proper subset
$S_{G}\in V(G)\cup E(G)$ corresponds another proper subset $S_{H}\in V(H)\cup
E(H)$ such that $G-S_{G}\cong H-S_{H}$, then $G\cong H$.

######  Conjecture 2.

∗ Let both $\mathcal{H}_{yper}=(\Lambda,\mathcal{E})$ and
$\mathcal{H}^{*}_{yper}=(\Lambda^{*},\mathcal{E}^{*})$ be hypergraphs with
$|\mathcal{E}|=|\mathcal{E}^{*}|$. If there is a bijection
$\theta:\mathcal{E}\rightarrow\mathcal{E}^{*}$ such that two partial
hypergraphs
$\mathcal{E}\setminus\\{e\\}\cong\mathcal{E}^{*}\setminus\\{\theta(e)\\}$ for
each hyperedge $e\in\mathcal{E}$, then
$\mathcal{H}_{yper}\cong\mathcal{H}^{*}_{yper}$.

####  5.2.3 Set-cut-sets, Hypergraph connectivity

By the vertex-splitting operation defined in Definition 1, we vertex-split
each vertex $w$ in a non-empty vertex subset $S$ of a _hyperedge connected
intersected-graph_ $G$ into two vertices $w\,^{\prime}$ and
$w\,^{\prime\prime}$, such that $N(w)=N(w\,^{\prime})\cup
N(w\,^{\prime\prime})$ with $N(w\,^{\prime})\cap
N(w\,^{\prime\prime})=\emptyset$, $|N(w\,^{\prime})|\geq 1$ and
$|N(w\,^{\prime\prime})|\geq 1$, the resultant graph is denoted as $G\wedge
S$, and let $S\,^{\prime}=\\{w\,^{\prime}:w\in S\\}$ and
$S\,^{\prime\prime}=\\{w\,^{\prime\prime}:w\in S\\}$, so $V(G\wedge
S)=V(G-S)\cup S\,^{\prime}\cup S\,^{\prime\prime}$. Since $G$ is the hyperedge
connected intersected-graph of a hyperedge connected hypergraph
$\mathcal{H}_{yper}=(\Lambda,\mathcal{E})$ (refer to Definition 21), so it
admits a set-coloring $F:V(G)\rightarrow\mathcal{E}$. We define a set-coloring
$F^{*}$ of the vertex-split graph $G\wedge S$ as:

(A-1) $F^{*}(x)=F(x)$ for $x\not\in S\,^{\prime}\cup S\,^{\prime\prime}$ and
$F^{*}(w\,^{\prime})=F(w)$ and $F^{*}(w\,^{\prime\prime})=F(w)$ for $w\in S$
and $w\,^{\prime},w\,^{\prime\prime}\in S\,^{\prime}\cup S\,^{\prime\prime}$;
and

(A-2) $F^{*}(uv)=F(uv)$ for $uv\in E(G-S)$, $F^{*}(uw\,^{\prime})=F(uw)$ and
$F^{*}(uw\,^{\prime\prime})=F(uw)$ for edges $uw\,^{\prime}$,
$uw\,^{\prime\prime}\in E(G\wedge S)$.

Suppose that the vertex-split graph $G\wedge S$ has subgraphs
$G_{1},G_{2},\dots,G_{m}$, such that

(i) $V(G\wedge S)=\bigcup^{m}_{i=1}V(G_{i})$ with $V(G_{i})\cap
V(G_{j})=\emptyset$ if $i\neq j$, and

(ii) $E(G\wedge S)=\bigcup^{m}_{i=1}E(G_{i})$ with $E(G_{i})\cap
E(G_{j})=\emptyset$ if $i\neq j$,
then $G\wedge S$ is not hyperedge connected, also, $G\wedge S$ is disconnected
in general, and we call $S$ a _vertex-split set-cut-set_. For each subgraph
$G_{i}$, the set-coloring $F$ induces a set-coloring
$F_{i}:V(G_{i})\rightarrow\mathcal{E}_{i}$ with
$\mathcal{E}_{i}\subset\mathcal{E}$, and call each subgraph $G_{i}$ _partial
hypergraph_.

Conversely, we do the vertex-coinciding operation defined in Definition 1 to
the subgraphs $G_{1},G_{2},\dots,G_{m}$ of the vertex-split graph $G\wedge S$
by vertex-coinciding two vertices $w\,^{\prime}$ and $w\,^{\prime\prime}$ into
one vertex $w=w\,^{\prime}\odot w\,^{\prime\prime}$, and get the original
hyperedge connected intersected-graph $G$, we write this case as
$G=\odot\langle G_{1},G_{2},\dots,G_{m}\rangle$.

######  Definition 26.

∗ Hypergraph connectivity. Suppose that a vertex-split graph $G\wedge S^{*}$
of the hyperedge connected intersected-graph $G$ of a hypergraph
$\mathcal{H}_{yper}=(\Lambda,\mathcal{E})$ holds $|S^{*}|\leq|S|$ for any
vertex-split graph $G\wedge S$, where $G\wedge S$ is not hyperedge connected,
then the number $|S^{*}|$ is called the _hyperedge split-connect number_ ,
written as $n_{vsplit}(G)=n_{vsplit}(\mathcal{H}_{yper})$. Since $G-S^{*}$ is
a disconnected graph having components
$G_{1}-S^{*},G_{2}-S^{*},\dots,G_{m}-S^{*}$, that is, the hyperedge connected
intersected-graph $G$ is _vertex $|S^{*}|$-connectivity_, and the hypergraph
$\mathcal{H}_{yper}$ is _hyperedge $|\mathcal{E}^{*}|$-connectivity_, where
$\mathcal{E}^{*}=\\{e:F(w)=e\in\mathcal{E},w\in S^{*}\\}$ makes the hyperedge
set $\mathcal{E}\setminus\mathcal{E}^{*}$ to be _hyperedge disconnected_ , we
call the hyperedge set $\mathcal{E}^{*}$ a _hyperedge set-cut-set_ of the
hypergraph $\mathcal{H}_{yper}$. $\square$

######  Remark 6.

In the view of decomposition, the hyperedge connected intersected-graph $G$
can be decomposed into vertex-disjoint partial hypergraphs
$G_{1},G_{2},\dots,G_{m}$.

For $w\in S$ and $w\,^{\prime},w\,^{\prime\prime}\in S\,^{\prime}\cup
S\,^{\prime\prime}$ in (A-1) above, we redefine $F^{*}(w\,^{\prime})=F(uw)$
and $F^{*}(w\,^{\prime\prime})=F(w)\setminus F^{*}(w\,^{\prime})$, since
$F(uw)\subseteq F(w)\cap F(u)$. Thereby, we get more families of subgraphs
like $G_{1},G_{2},\dots,G_{m}$, in other words, a hyperedge connected
hypergraph can be decomposed into many groups of hyperedge disjoint partial
hypergraphs. $\parallel$

As known, the vertex-splitting connectivity of a connected graph is equivalent
to its own vertex connectivity (Ref. [38]), so we have

######  Theorem 37.

The hyperedge split-connected number of a hyperedge connected hypergraph is
equal to its own hyperedge connectivity.

###  5.3 Miscellaneous set-type colorings

####  5.3.1 Pan-operation graphs

######  Definition 27.

∗ Let $(\bullet)$ be an operation on sets, and let $\mathcal{E}$ be a
hyperedge set defined on a finite set $\Lambda$ such that each hyperedge
$e\in\mathcal{E}$ corresponds another hyperedge $e\,^{\prime}\in\mathcal{E}$
holding $e(\bullet)e\,^{\prime}$ to be one element of the power set
$\Lambda^{2}$, and $\Lambda=\bigcup_{e\in\mathcal{E}}e$, as well as
$R_{est}(m+1)=\\{c_{0},c_{1},c_{2},\dots,c_{m}\\}=\\{c_{k}\\}^{m}_{k=0}$ be a
_constraint set_ with $m\geq 0$, in which
$c_{0}:=e(\bullet)e\,^{\prime}\in\Lambda^{2}$. If a graph $G$ admits a proper
set-coloring $\theta:V(H)\rightarrow\mathcal{E}$ holding
$\theta(x)\neq\theta(y)$ for each edge $xy\in E(G)$, and there is a function
$\theta$ such that each edge $uv$ of $E(G)$ is colored with
$\theta(uv)=\pi(\theta(u),\theta(v))$, and

(i) $c_{0}:~{}\theta(uv)\supseteq\theta(u)(\bullet)\theta(v)\neq\emptyset$.

(ii) For some $k\in[1,m]$, there is $c_{k}:=\pi(b_{u},b_{uv},b_{v})$ for
$b_{uv}\in\theta(uv)$, $b_{u}\in\theta(u)$ and $b_{v}\in\theta(v)$.

(iii) Any pair of hyperedges $e_{i},e_{j}\in\mathcal{E}$ holding
$e_{i}(\bullet)e_{j}\in\Lambda^{2}$ corresponds an edge $x_{i}y_{j}$ of $G$,
such that $\theta(x_{i})=e_{i}$ and $\theta(y_{j})=e_{j}$.

We call $G$ _pan-operation graph_ of the hypergraph
$\mathcal{H}_{yper}=(\Lambda,\mathcal{E})$. $\square$

######  Example 10.

Let $\mathcal{E}$ be a hyperedge set defined on $\Lambda=[1,q]$. For a fixed
integer $k>0$, there exists $c_{i,j}\in e_{i,j}\in\mathcal{E}$, such that the
operation $e_{i}(\bullet)e_{j}:=\varphi(a_{i},c_{i,j},b_{j})=k$ for some
$a_{i}\in e_{i}$ and $b_{j}\in e_{j}$, where $\varphi(a_{i},c_{i,j},b_{j})=k$
is one of $a_{i}+c_{i,j}+b_{j}=k$, $c_{i,j}+|a_{i}-b_{j}|=k$,
$|a_{i}+b_{j}-c_{i,j}|=k$ and $\big{|}|a_{i}-b_{j}|-c_{i,j}\big{|}=k$, refer
to Definition 8.

For a fixed integer $k>0$, another hyperedge set $\mathcal{E}^{*}$ defined on
$\Lambda=[1,q]$ holds $|e|\geq 3$ for each hyperedge $e\in\mathcal{E}^{*}$. We
have sets $\varphi(e,k)=\\{a_{i},b_{i},c_{i}\in
e:\varphi(a_{i},c_{i},b_{i})=k\\}$ for each $e\in\mathcal{E}^{*}$. Each
hyperedge $e\in\mathcal{E}^{*}$ corresponds another hyperedge
$e\,^{\prime}\in\mathcal{E}^{*}$, such that the operation
$e_{i}(\bullet)e_{j}$ to be
$\varphi(e,k)\cap\varphi(e\,^{\prime},k)\neq\emptyset$. $\square$

######  Example 11.

Let $G$ be a bipartite $(p,q)$-graph with vertex set bipartition $(X,Y)$, and
let $\mathcal{E}$ be a hyperedge set defined on $\Lambda=[1,q]$. If $G$ admits
a set-coloring $F:V(G)\rightarrow\mathcal{E}$, such that

$\max\\{\max F(x):x\in X\\}<\min\\{\max F(y):y\in Y\\}$ (42)

And the operation $e_{i}(\bullet)e_{j}$ means that each edge $xy\in E(G)$
corresponds some hyperedge $e_{x,y}\in\mathcal{E}$ containing a number
$c_{x,y}=a_{y}-b_{x}>0$ for some numbers $a_{y}\in F(y)$ and $b_{x}\in F(x)$.
Moreover, if the set $\\{c_{x,y}\in e_{x,y}:xy\in
E(G),e_{x,y}\in\mathcal{E}\\}=[1,q]$, we call $F$ a _set-ordered e-graceful
set-coloring_ based on the hyperedge set $\mathcal{E}$. $\square$

######  Definition 28.

[29] Let $G_{1},G_{2},\dots,G_{p+q}$ be the copies of a $(p,q)$-graph $G$
admitting a magic total labeling $h:V(G)\cup E(G)\rightarrow[1,p+q]$ such that
$h(u)+h(v)=k+h(uv)$ for each edge $uv\in E(G)$. Each $G_{i}$ with
$i\in[1,p+q]$ admits a magic total labeling $h_{i}:V(G_{i})\cup
E(G_{i})\rightarrow[1,p+q]$ holding

$h_{i}(x)+h_{i}(y)=k_{i}+h_{i}(xy)~{}(\textrm{mod}~{}p+q)$ (43)

true for each edge $xy\in E(G_{i})$, and the index set
$\\{k_{1},k_{2},\dots,k_{p+q}\\}=[1,p+q]$, and we write
$F_{p+q}(G,h)=\\{G_{1},G_{2}$, $\dots$, $G_{p+q}\\}$. For any _preappointed
zero_ $G_{k}\in F_{p+q}(G,h)$, we have

$h_{i}(w)+h_{i}(w)-h_{k}(w)=h_{\lambda}(w)$ (44)

with $\lambda=i+j-k~{}(\bmod~{}p+q)$ for each $w\in V(G)\cup E(G)$, and we
call the set $F_{p+q}(G,h)$ an _every-zero total graphic group_ , and rewrite
it as $\\{F_{p+q}(G,h);\oplus\\}$. $\square$

######  Definition 29.

[32] An _every-zero graphic group_ $\\{F_{n}(H);\oplus\\}$ made by a Topsnut-
gpw $H$ admitting a $\varepsilon$-labeling $h$ contains its own elements
$H_{i}\in F_{n}(H)=\\{H_{i}:i\in[1,n]\\}$ with $n\geq q$ ($n\geq 2q-1$)
holding $H\cong H_{i}$ and admitting a $\varepsilon$-labeling $h_{i}$ induced
by $h$ with $i\in[1,n]$ and hold an additive operation $H_{i}\oplus H_{j}$
defined as

$h_{i}(x)+h_{j}(x)-h_{k}(x)=h_{\lambda}(x)$ (45)

with $\lambda=i+j-k\,(\bmod\,n)$ for each element $x\in V(H)$ under any
_preappointed zero_ $H_{k}$. $\square$

######  Example 12.

Refer to Definition 28 and Definition 29. Let $\\{F(G);\oplus\\}$ be an
_every-zero graphic group_ based on a graphic set
$F(G)=\\{G_{1},G_{2},\dots,G_{q}\\}$ obtained by a $(p,q)$-graph $G_{1}$
admitting a graceful labeling $f_{1}$, where each graph $G_{i}$ holds
$G_{i}\cong G_{1}$ and admits a labeling $f_{i}$ defined as
$f_{i}(x)=f_{1}(x)+i~{}(\bmod~{}q)$ for $x\in V(G_{i})=V(G_{1})$ and
$f_{i}(xy)=|f_{i}(x)-f_{i}(y)|$ for $xy\in E(G_{i})=E(G_{1})$ with
$i\in[1,q]$. Selecting arbitrarily any graph $G_{k}\in F(G)$ as the
_preappointed zero_ , the additive operation $G_{i}\oplus G_{j}$ is defined as

$f_{i}(x)+f_{j}(x)-f_{k}(x)=f_{\beta}(x)$ (46)

for $x\in V(G_{i})=V(G_{j})=V(G_{k})=V(G_{1})$ and
$\beta=i+j-k~{}(\bmod~{}q)$. It is not hard to show that $\\{F(G);\oplus\\}$
holds Zero element, Inverse, Uniqueness, Closure and Associative law.

Let $\Lambda=F(G)$, and let $\mathcal{E}$ be a hyperedge set defined on
$\Lambda$.

Case 1. Suppose that a $(p\,^{\prime},q\,^{\prime})$-graph $H$ admits an _e-
index-graceful set-labeling_ $F:V(H)\rightarrow\mathcal{E}$, and the induced
edge color $F(uv)=\\{G_{\beta}\\}$ obtained by $G_{i}\oplus G_{j}=G_{\beta}$
with $G_{i}\in F(u)$ and $G_{j}\in F(v)$ and $\beta=i+j-k~{}(\bmod~{}q)$, such
that the index set $\\{r:\\{G_{r}\\}=F(uv),uv\in E(H)\\}=[1,q\,^{\prime}]$. It
is noticeable, $G_{i}\oplus G_{j}$ means the operation $e_{i}(\bullet)e_{j}$
defined in Definition 27, since $e_{i}=F(u)$ and $e_{j}=F(v)$.

Case 2. Suppose that a $(p\,^{\prime},q\,^{\prime})$-graph $H$ admits a set-
labeling $F^{*}:V(H)\cup E(H)\rightarrow\mathcal{E}$, for each edge $uv\in
E(H)$, there are $G_{i}\in F^{*}(u)$, $G_{j}\in F^{*}(v)$ and $G_{\beta}\in
F^{*}(uv)$ holding $G_{i}\oplus G_{j}=G_{\beta}$ with
$\beta=i+j-k~{}(\bmod~{}q)$, Here, the operation $e_{i}(\bullet)e_{j}$ defined
in Definition 27 is that $G_{i}\oplus G_{j}=G_{\beta}\in F^{*}(uv)$ based on
$e_{i}=F(u)$ and $e_{j}=F(v)$. However, $F(uv)\neq F^{*}(uv)$ for each edge
$uv\in E(H)$ in general, where $F$ is defined in Case 1. $\square$

######  Example 13.

Let $S_{pan}(K_{n})$ be the set of spanning trees of a complete graph $K_{n}$
admitting a labeling defined on $[1,n]$, so the cardinality
$|S_{pan}(K_{n})|=n^{n-2}$ by the famous Cayley’s formula. We have a hyperedge
set $\mathcal{E}$ with elements $e_{i}=E(T_{i})$ for $T_{i}\in
S_{pan}(K_{n})$. Thereby,
$\Lambda=E(K_{n})=\bigcup_{e_{i}\in\mathcal{E}}e_{i}$. If two spanning trees
$T_{i}$ and $T_{j}$ hold $T_{j}=T_{i}+x_{i}y_{i}-u_{i}v_{i}$ for
$x_{i}y_{i}\not\in E(T_{i})$ and $u_{i}v_{i}\in E(T_{i})$, we write this fact
by $e_{i}[+,-]e_{j}$. A graph $G^{*}$ admits a set-coloring
$F^{*}:V(G^{*})\rightarrow\mathcal{E}$, such that $F^{*}(w)\neq F^{*}(t)$ for
distinct vertices $w,t\in V(G^{*})$, and each edge $uv$ is colored with an
induced color set $F^{*}(uv)=F^{*}(u)[+,-]F^{*}(v)$, and each vertex degree
$\textrm{deg}_{G^{*}}(w)\geq n-1$, and $G^{*}$ has $n^{n-2}$ vertices in
total. Here, “$(\bullet)$”=“$[+,-]$”.

We select spanning trees $T^{c}_{1},T^{c}_{2},\dots,T^{c}_{m}$ from the set
$S_{pan}(K_{n})$ to form a base

$\textbf{{T}}^{c}=(T^{c}_{1},T^{c}_{2},\dots,T^{c}_{m})$

and have a permutation $J_{1},J_{2},\dots,J_{A}$ of spanning trees
$a_{1}T^{c}_{1},a_{2}T^{c}_{2},\dots,a_{m}T^{c}_{m}$, where
$A=\sum^{m}_{k=1}a_{k}$. We do the operation “$[+,-]$” to $J_{1}$ and $J_{2}$,
and then get a spanning tree $H_{1}=J_{1}[+,-]J_{2}$, next we get another
spanning tree $H_{2}=H_{1}[+,-]J_{3}$, go on in this way, we get spanning
trees $H_{k}=H_{k-1}[+,-]J_{k+1}$ for $k\in[1,A]$ with $H_{0}=J_{1}$. Hence,
we write $H_{A}=[+,-]^{m}_{k=1}\langle a_{k}T^{c}_{k}\rangle$, and call the
following set

$\textbf{{L}}(Z^{0}[+,-]\textbf{{T}}^{c})=\left\\{[+,-]^{m}_{k=1}\langle
a_{k}T^{c}_{k}\rangle:~{}a_{k}\in Z^{0},T^{c}_{k}\in\textbf{{T}}^{c}\right\\}$
(47)

_K-spanning tree lattice_. $\square$

####  5.3.2 Colorings of hypergraphs

######  Definition 30.

∗ Let $\mathcal{E}$ be a set of subsets of a finite set $\Lambda$ such that
each hyperedge $e\in\mathcal{E}$ satisfies $e\neq\emptyset$ and corresponds
some hyperedge $e\,^{\prime}\in\mathcal{E}$ holding $e\cap
e\,^{\prime}\neq\emptyset$, as well as $\Lambda=\bigcup_{e\in\mathcal{E}}e$.
Suppose that a connected graph $H$ admits an _edge set-labeling_
$F\,^{\prime}:E(H)\rightarrow\mathcal{E}$ holding $F\,^{\prime}(uv)\neq
F\,^{\prime}(uw)$ for any two adjacent edges $uv,uw\in E(H)$, and the vertex
color set $F\,^{\prime}(w)$ for each vertex $w\in V(G)$ is induced by one of
the following riles:

  1. Edgeinduce-1.

$F\,^{\prime}(w)=\\{F\,^{\prime}(wz):z\in N(w)\\}\subseteq\Lambda^{2}$.

  2. Edgeinduce-2.

$F\,^{\prime}(w)=\bigcup_{z\in N(w)}F\,^{\prime}(wz)\subseteq\Lambda$.

We call $H$ an _edge-set-colored graph_. The _edge-induced graph_ $L_{H}$ of
the edge-set-colored graph $H$ has its own vertex set $V(L_{H})=E(H)$, and
admits a vertex set-coloring $F\,^{\prime}:V(L_{H})\rightarrow\mathcal{E}$,
such that two vertices $w_{uv}(:=uv)$ and $w_{xy}(:=xy)$ of $V(L_{H})$ are the
ends of an edge of $L_{H}$ if and only if $F\,^{\prime}(w_{uv})\cap
F\,^{\prime}(w_{xy})\neq\emptyset$ (i.e. $F\,^{\prime}(uv)\neq
F\,^{\prime}(xy)$ in $H$). So, this edge-induced graph $L_{H}$ is the
_intersected-graph_ of a hypergraph
$\mathcal{H}_{yper}=(\Lambda,\mathcal{E})$. $\square$

######  Theorem 38.

A proper hyperedge coloring of a hypergraph
$\mathcal{H}_{yper}=(\Lambda,\mathcal{E})$ is equivalent to a proper vertex-
coloring of the intersected-graph $G$ of this hypergraph $\mathcal{H}_{yper}$,
and vice versa. Thereby, we have
$\chi(G)=\chi\,^{\prime}(\mathcal{H}_{yper})$, where $\chi(G)$ is the
_chromatic number_ of $G$, and $\chi\,^{\prime}(\mathcal{E})$ is the
_hyperedge chromatic index_ of $\mathcal{H}_{yper}$.

######  Remark 7.

In Definition 30, an edge-set-colored graph $H$ may be a subgraph of the
intersected-graph of a hypergraph
$\mathcal{H}_{yper}=(\Lambda,\mathcal{E}^{*})$ with
$\mathcal{E}^{*}\neq\mathcal{E}$ because of Edgeinduce-1 and Edgeinduce-2
defined in Definition 30. The _edge-induced graph_ $L_{H}$ is not the _line
graph_ of the edge-set-colored graph $H$.

Theorem 38 tells us that the proper hyperedge coloring problem of a hypergraph
is a NP-problem, since there is a well-known conjecture in the proper vertex-
coloring of graphs, that is, Bruce Reed in 1998 conjectured: The _chromatic
number_ $\chi(G)\leq\left\lceil\frac{\Delta(G)+1+K(G)}{2}\right\rceil$, where
$\Delta(G)$ is the _maximum degree_ of $G$ and $K(G)$ is the _maximum clique
number_ of $G$. $\parallel$

######  Problem 15.

Colorings of hypergraphs: How to color the vertices of the hypergraph
$\mathcal{H}_{yper}=(\Lambda,\mathcal{E})$ such that no edge of $\mathcal{E}$
has all its vertices with the same color.

######  Remark 8.

For a given hyperedge set $\mathcal{E}$, we color the vertices of the vertex
set $\Lambda$ with $k$ colors such that each set $e\in\mathcal{E}$ contains
two vertices colored with different colors if the cardinality $|e|\geq 2$.
Clearly, different hyperedge sets correspond different vertex-colorings of the
vertex set $\Lambda$. The number

$\chi(\Lambda,\mathcal{E})=\min\\{k:\textrm{ each $k$-coloring of $\Lambda$
based on a hyperedge set $\mathcal{E}$}\\}$ (48)

is called the _hypervertex chromatic number_ of the hypergraph
$\mathcal{H}_{yper}=(\Lambda,\mathcal{E})$. $\parallel$

######  Definition 31.

∗ A _hyper-total coloring_ $\theta$ of a hypergraph
$\mathcal{H}_{yper}=(\Lambda,\mathcal{E})$ is defined by

(i) $\theta:\mathcal{E}\rightarrow[1,b]$, and $\theta(e_{i})\neq\theta(e_{j})$
if $e_{i}\cap e_{j}\neq\emptyset$;

(ii) $\theta(x_{i,j})\in[a,b]$, and $\theta(x_{i,j})\neq\theta(x_{i,k})$ for
some distinct $x_{i,j},x_{i,k}\in e_{i}$ if $|e_{i}|\geq 2$.
And $\chi\,^{\prime\prime}(\Lambda,\mathcal{E})$ is the smallest number of $b$
for which $\mathcal{H}_{yper}$ admits a hyper-total coloring.

A _hyperedge coloring_ $\varphi:\mathcal{E}\rightarrow[1,M]$, such that the
elements of hyperedge set

$N(e_{i})=\big{\\{}e_{j}:e_{i}\cap
e_{j}\neq\emptyset,e_{j}\in\mathcal{E}\setminus\\{e_{i}\\}\big{\\}}$

are colored different colors from each other, and the number
$\max\\{|N(e_{i})|:e_{i}\in\mathcal{E}\\}=\Delta(\mathcal{E}_{\cap})$ holds
the following inequalities

$\Delta(\mathcal{E}_{\cap})\leq M\leq\Delta(\mathcal{E}_{\cap})+1$ (49)

true by the famous Vizing’s theorem on the edge coloring of the intersected-
graph of the hypergraph $\mathcal{H}_{yper}$. $\square$

####  5.3.3 The set of hyperedge sets

######  Definition 32.

∗ Let $H^{yper}_{color}(G)$ be the set of hyperedge sets of all set-colorings
of a connected graph $G$, such that each hyperedge set $\mathcal{E}\in
H^{yper}_{color}(G)$ defines a set-coloring $F:V(G)\rightarrow\mathcal{E}$,
where $\bigcup_{e\in\mathcal{E}}e=\Lambda$, and $G$ is the intersected-graph
of a hypergraph $\mathcal{H}_{yper}=(\Lambda,\mathcal{E})$. For another set-
coloring $F\,^{\prime}:V(G)\rightarrow\mathcal{E}\,^{\prime}\in
H^{yper}_{color}(G)$ with
$\bigcup_{e\in\mathcal{E}\,^{\prime}}e=\Lambda\,^{\prime}$, we defined a third
set-coloring $F^{*}:V(G)\rightarrow\mathcal{E}^{*}$, such that
$F^{*}(x)=F(x)\cup F\,^{\prime}(x)$ for each vertex $x\in V(G)$ and
$F^{*}(uv)=F(uv)\cup F\,^{\prime}(uv)$ for each edge $uv\in E(G)$. Clearly,
$\mathcal{E}^{*}=\mathcal{E}\cup\mathcal{E}\,^{\prime}$, and

(i) $\Lambda^{*}=\Lambda=\Lambda\,^{\prime}$; or

(ii) $\Lambda^{*}=\Lambda\cup\Lambda\,^{\prime}$ if
$\Lambda\neq\Lambda\,^{\prime}$.
We call the set-coloring $F^{*}$ _union set-coloring_ of two set-colorings $F$
and $F\,^{\prime}$, the hyperedge set $F^{*}$ _hyperedge union set_ of two
hyperedge sets $\mathcal{E}$ and $\mathcal{E}\,^{\prime}$. $\square$

######  Problem 16.

The set $H^{yper}_{color}(G)$ defined in Definition 32 can be classified into
two parts

$H^{yper}_{color}(G)=I^{yper}_{color}(G)\bigcup N^{yper}_{color}(G)$ (50)

such that each hyperedge set of $I^{yper}_{color}(G)$ is not the union set-
coloring of any two set-colorings of the set $H^{yper}_{color}(G)$, but each
hyperedge set of $N^{yper}_{color}(G)$ is the hyperedge union set of some two
hyperedge sets of $H^{yper}_{color}(G)$. Determine the hyperedge set
$I^{yper}_{color}(G)$ for a connected graph $G$.

####  5.3.4 Set-set-colorings, Compound hypergraphs

From studying relationship between communities in networks, which is the
topological structure between hypergraphs, we propose the following two
concepts of set-set-coloring and compound hypergraph:

######  Definition 33.

∗ A graph $G$ admits a proper _set-set-coloring_
$\theta:V(G)\rightarrow\\{\mathcal{S}_{i}\\}^{n}_{i=1}$ with
$\theta(x)\neq\theta(y)$ for each edge $xy\in E(G)$, where each
$\mathcal{S}_{i}$ is a set of subsets of $\Lambda^{2}$, such that each induced
edge color set is defined as
$\theta(u_{i}v_{j})=\theta(u_{i})(\bullet)\theta(v_{j})=\mathcal{S}_{i}(\bullet)\mathcal{S}_{j}$
subject to a restrictive condition set $R(m)$ based on an abstract operation
“$(\bullet)$”. In particularly, each edge is colored with an induced set

$\theta(u_{i}v_{j})\supseteq\theta(u_{i})\cap\theta(v_{j})=\mathcal{S}_{i}\cap\mathcal{S}_{j}\neq\emptyset$

when “$(\bullet)$” $=$ “$\bigcap$” is the intersection of sets. $\square$

######  Definition 34.

∗ Suppose that a graph $G$ admits a proper _compound set-coloring_
$\Gamma:V(G)\rightarrow\\{\mathcal{E}_{i}\\}^{n}_{i=1}$ with
$\Gamma(x)\neq\Gamma(y)$ for each edge $xy\in E(G)$, where each
$\mathcal{E}_{i}$ is a set of subsets of a finite set $\Lambda$, and
$\Lambda=\bigcup^{n}_{i=1}\Lambda_{i}$ with
$\Lambda_{i}=\bigcup_{e_{i,j}\in\mathcal{E}_{i}}e_{i,j}$, and each
$(\Lambda_{i},\mathcal{E}_{i})$ is a hypergraph. If there are a function
$\psi$ and a restrictive condition set $R(m)$ such that each edge $u_{i}v_{j}$
is colored with an induced edge color set

$\Gamma(u_{i}v_{j})=\psi(\Gamma(u_{i}),\Gamma(v_{j}))\supseteq\Gamma(u_{i})\cap\Gamma(v_{j})=\mathcal{E}_{u_{i}}\cap\mathcal{E}_{u_{j}}\neq\emptyset$
(51)

or there is an abstract operation “$(\bullet)$” such that each induced edge
color set

$\Gamma(u_{i}v_{j})=\Gamma(u_{i})(\bullet)\Gamma(v_{j})=\mathcal{E}_{u_{i}}(\bullet)\mathcal{E}_{u_{j}}\neq\emptyset$
(52)

subject to $R(m)$, then we call
$\mathcal{H}_{comp}=\big{(}\Lambda,\bigcup^{n}_{i=1}\mathcal{E}_{i}\big{)}$
_compound hypergraph_ , and $G$ the _compound intersected-graph_ of the
compound hypergraph $\mathcal{H}_{comp}$ if each
$\mathcal{E}_{u_{i}}\cap\mathcal{E}_{u_{j}}\neq\emptyset$ corresponds an edge
$u_{i}v_{j}$ of the graph $G$. $\square$

######  Example 14.

Let $K_{2n}$ be a complete graph of $2n$ vertices. Then $K_{2n}$ has perfect
matching groups $\textbf{{M}}_{i}=\\{M_{i,1},M_{i,2}$, $\dots$,
$M_{i,2n-1}\\}$ for $i\in[1,m]$. There is a graph $G$ admitting a set-coloring
$F:V(G)\rightarrow\mathcal{E}_{i}$ with each hyperedge set
$\mathcal{E}_{i}=\textbf{{M}}_{i}$ such that $F(x)\neq F(y)$ for distinct
vertices $x,y\in V(G)$, and each edge $uv\in E(G)$ is colored with an induced
set $F(uv)=F(u)\cup F(v)$ if $E(C_{k})=F(u)\cup F(v)$, where $C_{k}$ is a
Hamilton cycle of $K_{2n}$. If $G$ is a complete graph, then it shows the
Perfect 1-Factorization Conjecture (Anton Kotzig, 1964):

“ _For any $n\geq 2$, $K_{2n}$ can be decomposed into $2n-1$ perfect matchings
such that the union of any two matchings forms a hamiltonian cycle of
$K_{2n}$_”.

Here, the union operation “$\bigcup$” is the abstract operation “$(\bullet)$”
appeared in Definition 34. $\square$

######  Definition 35.

∗ According to Definition 34, the compound intersected-graph $G$ admits a
compound set-coloring $\Gamma:V(G)\rightarrow\\{\mathcal{E}_{i}\\}^{n}_{i=1}$
holding Eq.(51), so $G$ corresponds its own _hypergraph-type Topcode-matrix_

${\begin{split}H^{comp}_{yper}(G)&=\left(\begin{array}[]{ccccc}\Gamma(x_{1})&\Gamma(x_{2})&\cdots&\Gamma(x_{q})\\\
\Gamma(x_{1}y_{1})&\Gamma(x_{2}y_{2})&\cdots&\Gamma(x_{q}y_{q})\\\
\Gamma(y_{1})&\Gamma(y_{2})&\cdots&\Gamma(y_{q})\end{array}\right)=\left(\begin{array}[]{ccccc}\mathcal{E}_{x_{1}}&\mathcal{E}_{x_{2}}&\cdots&\mathcal{E}_{x_{q}}\\\
\mathcal{E}_{x_{1}y_{1}}&\mathcal{E}_{x_{2}y_{2}}&\cdots&\mathcal{E}_{x_{q}y_{q}}\\\
\mathcal{E}_{y_{1}}&\mathcal{E}_{y_{2}}&\cdots&\mathcal{E}_{y_{q}}\end{array}\right)\\\
&=(X(\mathcal{E}),~{}E(\mathcal{E}),~{}Y(\mathcal{E}))^{T}\end{split}}$ (53)

where $E(G)=\\{x_{i}y_{i}:i\in[1,q]\\}$,
$X(\mathcal{E})=(\mathcal{E}_{x_{1}},\mathcal{E}_{x_{2}},\cdots,\mathcal{E}_{x_{q}})$
and
$Y(\mathcal{E})=(\mathcal{E}_{y_{1}},\mathcal{E}_{y_{2}},\cdots,\mathcal{E}_{y_{q}})$
are called _v-hypergraph vectors_ , and
$E(\mathcal{E})=(\mathcal{E}_{x_{1}y_{1}},\mathcal{E}_{x_{2}y_{2}},\cdots,\mathcal{E}_{x_{q}y_{q}})$
is called _e-hypergraph intersection vector_ , as well as
$\mathcal{E}_{x_{i}y_{i}}=\psi(\mathcal{E}_{x_{i}},\mathcal{E}_{y_{i}})$ for
each edge $x_{i}y_{i}\in E(G)$. $\square$

######  Remark 9.

The matrix $H^{comp}_{yper}(G)$ defined in Eq.(53) in Definition 35 is a three
dimensional matrix, and the compound intersected-graph $G$ has its own
vertices as hypergraphs, its own edges as intersections of hypergraphs.
$\parallel$

###  5.4 Set-colored graph homomorphisms, Hypergraph homomorphisms

Let $G$ be a set-colored graph admitting a set-coloring
$F:V(G)\rightarrow\mathcal{E}$, where the hyperedge set $\mathcal{E}$ is a set
of subsets of a finite set $\Lambda$, and $H$ be a set-colored graph admitting
a set-coloring $F^{*}:V(H)\rightarrow\mathcal{E}^{*}$, where the hyperedge set
$\mathcal{E}^{*}$ is a set of subsets of a finite set $\Lambda^{*}$.

Suppose that $G$ is _graph homomorphism_ to $H$, that is, $G\rightarrow H$ if
there is a mapping $\varphi:V(G)\rightarrow V(H)$ such that
$\varphi(u)\varphi(v)\in E(H)$ for each edge $uv\in E(G)$, refer to Definition
2. Then we get $\varphi(F(uv))=F^{*}(\varphi(u)\varphi(v))$, and

$F^{*}(w)=\varphi(F(x_{1}))\cup\varphi(F(x_{2}))\cup\cdots\cup\varphi(F(x_{k}))$

if $\varphi(x_{i})=w\in V(H)$ and $x_{i}\in V(G)$ with $i\in[1,k]$. We write
this _set-colored graph homomorphism_ as

$G[\mathcal{E}]\rightarrow H[\mathcal{E}^{*}]$ (54)

which induces a _hypergraph homomorphism_

$(\Lambda,\mathcal{E})\rightarrow(\Lambda^{*},\mathcal{E}^{*})$ (55)

as $\varphi:\Lambda\rightarrow\Lambda^{*}$.

######  Remark 10.

See Fig.26 and Fig.27, we have four set-colored graph homomorphisms
$S_{k}[\mathcal{E}_{k}]\rightarrow S_{k-1}[\mathcal{E}_{k-1}]$ for
$k\in[1,4]$, where the set-colored graph $S_{0}=G$ shown in Fig.8(b) is the
intersected-graph of a $8$-uniform hypergraph shown in Fig.8 (a). Conversely,
each set-colored graph $S_{i}$ is a result of doing the vertex-splitting
operation to $S_{i-1}$ with $i\in[1,4]$. More or less, we show the hyperedge-
splitting operation and the hyperedge-coinciding operation of hypergraphs.
$\parallel$

Figure 26: The first scheme for the set-colored graph homomorphism operation.
Figure 27: The second scheme for the set-colored graph homomorphism operation.

###  5.5 Set-colored graphic groups

######  Definition 36.

If a graph set $F_{\mathcal{E}}(G)=\\{G_{1},G_{2},\dots,G_{n}\\}$ holds:

(i) Each graph $G_{i}$ admits a set-coloring
$F_{i}:V(G_{i})\rightarrow\mathcal{E}_{i}$, where $\mathcal{E}_{i}$ is a
hyperedge set defined on an integer set $\Lambda_{i}$ for $i\in[1,n]$;

(ii) each graph $G_{i}$ is isomorphic to $G_{1}$, i.e. $G_{i}\cong G_{1}$;

(iii) there is a positive integer $M$, such that the color
$F_{i}(w_{s})=\\{b_{i,s,1},b_{i,s,2},\dots,b_{i,s,c(i,s)}\\}$ of each element
$w_{s}$ of $V(G_{i})\cup E(G_{i})$ is defined as
$b_{i,s,r}=b_{i,s,r}+i~{}(\bmod~{}M)$ with $r\in[1,c(i,j)]$ and $s\in[1,n]$,
where $M=\max\Lambda_{1}$; and

(iv) an additive operation “$G_{i}\oplus_{k}G_{j}$” defined by

$b_{i,s,r}+b_{j,s,r}-b_{k,s,r}=b_{\lambda,s,r}$ (56)

with $\lambda=i+j-k~{}(\bmod~{}M)$ for $b_{i,s,r}\in F_{i}(w_{s})$,
$b_{j,s,r}\in F_{j}(w_{s})$ and $b_{\lambda,s,r}\in F_{\lambda}(w_{s})$, as
well as $b_{k,s,r}\in F_{k}(w_{s})$, where $G_{k}$ is a pre-designated _zero_.

We call $F_{\mathcal{E}}(G)$ an _every-zero set-colored graphic group_ ,
rewrite it as $\\{F_{\mathcal{E}}(G);\oplus\\}$. $\square$

An every-zero set-colored graphic group is shown in Fig.28. We show the
following proof for the every-zero set-colored graphic group
$\\{F_{\mathcal{E}}(G);\oplus\\}$ defined in Definition 36:

Zero. Any colored graph $G_{k}\in\\{F_{\mathcal{E}}(G);\oplus\\}$ can be as
the _zero_ , so that
$G_{i}\oplus_{k}G_{k}=G_{i}\in\\{F_{\mathcal{E}}(G);\oplus\\}$ according to
Eq.(56).

Uniqueness and closure. If $G_{i},G_{j}\in\\{F_{\mathcal{E}}(G);\oplus\\}$
hold $G_{i}\oplus_{k}G_{j}=G_{s}$ and $G_{i}\oplus_{k}G_{j}=G_{r}$, then
$G_{s}=G_{r}\in\\{F_{\mathcal{E}}(G);\oplus\\}$ by Eq.(56). Closure stands up
by Eq.(56).

Inverse. The inverse $G_{i^{-1}}$ of each colored graph $G_{i}$ holds
$k=i+i^{-1}-k~{}(\bmod~{}M)$ for one of $i^{-1}=2k-i$ and $i^{-1}=M+2k-i$.

Associative law. Any three elements of $\\{F_{\mathcal{E}}(G);\oplus\\}$
satisfy

$(G_{i}\oplus_{k}G_{j})\oplus_{k}G_{r}=G_{i}\oplus_{k}(G_{j}\oplus_{k}G_{r})$

Commutative law. Any pair of two elements of $\\{F_{\mathcal{E}}(G);\oplus\\}$
hold $G_{i}\oplus_{k}G_{j}=G_{j}\oplus_{k}G_{i}$ true.

Figure 28: An every-zero set-colored graphic group for understanding
Definition 36.

######  Theorem 39.

∗ Suppose that $\\{F_{\mathcal{E}}(G);\oplus\\}$ is an every-zero set-colored
graphic group defined on a set-colored graph $G$ admitting a set-coloring
$F:V(G)\rightarrow\mathcal{E}$, where $\mathcal{E}$ is a hyperedge set defined
on an integer finite set $\Lambda$, then

(i) Each graph $G_{i}\in\\{F_{\mathcal{E}}(G);\oplus\\}$ admitting a set-
coloring $F_{i}:V(G)\rightarrow\mathcal{E}_{i}$ is the intersected-graph of
some hypergraph if $G$ is the intersected-graph of the hypergraph
$\mathcal{H}_{yper}=(\Lambda,\mathcal{E})$.

(ii) Each hyperedge set $\mathcal{E}_{i}$ contains a perfect hypermatching if
the hyperedge set $\mathcal{E}$ contains a perfect hypermatching.

(iii) Each graph $G_{i}\in\\{F_{\mathcal{E}}(G);\oplus\\}$ contains a
hyperedge path (res. hyperedge cycle) if $G$ contains a hyperedge path (res.
hyperedge cycle).

(iv) Each graph $G_{i}\in\\{F_{\mathcal{E}}(G);\oplus\\}$ is set-colored graph
homomorphism to a set-colored graph $H_{i}$ if $G$ is set-colored graph
homomorphism to $H$, so that $H_{i}\cong H$.

######  Definition 37.

∗ Suppose that a $(p,q)$-graph $J$ admits a total mapping $\phi:V(J)\cup
E(J)\rightarrow\\{F_{\mathcal{E}}(G);\oplus\\}$, where the very-zero set-
colored graphic group $\\{F_{\mathcal{E}}(G);\oplus\\}$ is defined in
Definition 36, such that each edge $xy$ holds $\phi(x)\neq\phi(y)$ and
$\phi(xy)=\phi(x)\oplus_{k}\phi(y)$ under a pre-designated zero
$G_{k}\in\\{F_{\mathcal{E}}(G);\oplus\\}$, and we get a _graph-type Topcode-
matrix_ of the $(p,q)$-graph $J$ as follows:

${\begin{split}T^{graph}_{code}(J)&=\left(\begin{array}[]{ccccc}\phi(u_{1})&\phi(u_{2})&\cdots&\phi(u_{q})\\\
\phi(u_{1}v_{1})&\phi(u_{2}v_{2})&\cdots&\phi(u_{q}v_{q})\\\
\phi(v_{1})&\phi(v_{2})&\cdots&\phi(v_{q})\end{array}\right)=\left(\begin{array}[]{ccccc}G_{u_{1}}&G_{u_{2}}&\cdots&G_{u_{q}}\\\
G_{u_{1}v_{1}}&G_{u_{2}v_{2}}&\cdots&G_{u_{q}v_{q}}\\\
G_{v_{1}}&G_{v_{2}}&\cdots&G_{v_{q}}\end{array}\right)\\\
&=(X(F_{\mathcal{E}}),~{}E(F_{\mathcal{E}}),~{}Y(F_{\mathcal{E}}))^{T}\end{split}}$
(57)

where $u_{i}v_{i}\in E(J)=\\{u_{i}v_{i}:i\in[1,q]\\}$. We call two vectors
$X(F_{\mathcal{E}})=(G_{u_{1}},G_{u_{2}},\cdots,G_{u_{q}})$ and
$Y(F_{\mathcal{E}})=(G_{v_{1}},G_{v_{2}},\cdots,G_{v_{q}})$ _v-graph vectors_
, and
$E(F_{\mathcal{E}})=(G_{u_{1}v_{1}},G_{u_{2}v_{2}},\cdots,G_{u_{q}v_{q}})$ _e-
graph vector_. Correspondingly, the above graph-type Topcode-matrix
$T^{graph}_{code}(J)$ of the $(p,q)$-graph $J$ induces a _matrix-type Topcode-
matrix_ defined as

${\begin{split}T^{matrix}_{code}(J)&=\left(\begin{array}[]{ccccc}T_{code}(G_{u_{1}})&T_{code}(G_{u_{2}})&\cdots&T_{code}(G_{u_{q}})\\\
T_{code}(G_{u_{1}v_{1}})&T_{code}(G_{u_{2}v_{2}})&\cdots&T_{code}(G_{u_{q}v_{q}})\\\
T_{code}(G_{v_{1}})&T_{code}(G_{v_{2}})&\cdots&T_{code}(G_{v_{q}})\end{array}\right)_{3\times
q}\\\ &=(X(T_{code}),~{}E(T_{code}),~{}Y(T_{code}))^{T}\end{split}}$ (58)

with an _e-Topcode-matrix vector_ $E(T_{code})=(T_{code}(G_{u_{1}v_{1}})$,
$T_{code}(G_{u_{2}v_{2}})$, $\cdots$, $T_{code}(G_{u_{q}v_{q}}))$, and two _v-
Topcode-matrix vectors_ $X(T_{code})=(T_{code}(G_{u_{1}})$,
$T_{code}(G_{u_{2}})$, $\cdots$, $T_{code}(G_{u_{q}}))$ and
$Y(T_{code})=(T_{code}(G_{v_{1}})$,
$T_{code}(G_{v_{2}}),\cdots,T_{code}(G_{v_{q}}))$. $\square$

######  Remark 11.

The matrix-type Topcode-matrix $T^{matrix}_{code}(J)$ defined in Eq.(58) in
Definition 37 is a _three dimensional matrix_ having the elements as Topcode-
matrices $T_{code}(G_{u_{i}})$, $T_{code}(G_{v_{i}})$ and
$T_{code}(G_{u_{i}v_{i}})$ of $3\times q$ rank, more or less, like some things
in _Tensor_. $\parallel$

###  5.6 Preliminarily studying of hypernetworks

Three physical scientists Newman, Barabási and Watts (Ref. [17]) pointed:“
_Pure graph theory is elegant and deep, but it is not especially relevant to
networks arising in the real world. Applied graph theory, as its name
suggests, is more concerned with real-world network problems, but its approach
is oriented toward design and engineering._ ”

Hypernetworks have been mentioned or studied by scholars for a long time.
However, there is no recognized definition of the hypernetwork, and some
articles consider hypernetworks as hypergraphs. Here, we will use the set-
coloring to obtain the _dynamic intersected-networks_ , so as to observe
hypernetworks, and try to plant some researching conclusions of dynamic
networks into hypernetworks.

####  5.6.1 Dynamic hypernetworks

######  Definition 38.

∗ At each time step $t\in[a,b]$, a _dynamic hypernetwork_
$\mathcal{N}_{yper}(t)=(\Lambda(t),\mathcal{E}(t))$ holds:

Hnet-1. Each hyperedge $e(t)\in\mathcal{E}(t)$ is not an empty set and
corresponds another hyperedge
$e\,^{\prime}(t)\in\mathcal{E}(t)\setminus\\{e(t)\\}$ to form $e(t)\cap
e\,^{\prime}(t)\neq\emptyset$;

Hnet-2. $\Lambda(t)=\bigcup_{e(t)\in\mathcal{E}(t)}e(t)$ is a finite set.
$\square$

Hereafter, we will omit “dynamic” from “dynamic hypernetwork” for the
simplicity of statement. In Definition 38, each hyperedge
$e(t)\in\mathcal{E}(t)$ describes an area network, community, blockchain,
logistics network, cybergroup, _etc_. Hypernetworks describe dynamic changes
among groups and communities in networks, and compound hypernetworks depict
correlations between two or more hypernetworks.

######  Definition 39.

∗ Suppose that $N_{int}(t)$ is the _intersected-network_ of a hypernetwork
$\mathcal{N}_{yper}(t)=(\Lambda(t),\mathcal{E}(t))$ defined in Definition 38
subject to a restrictive condition set $R_{est}(m+1)=\\{c_{0},c_{1},\dots$,
$c_{m}\\}$ with $m\geq 0$ at each time step $t\in[a,b]$, so the intersected-
network $N_{int}(t)$ admits a set-coloring
$F:V(N_{int}(t))\rightarrow\mathcal{E}(t)$, which induces edge color set
$F(uv)$ for each edge $uv\in E(N_{int}(t))$ such that

(i) the first restrictive condition $c_{0}$ holds $F(uv)\supseteq F(u)\cap
F(v)\neq\emptyset$ true; and

(ii)) for some $k\in[1,m]$, the $k$th constraint $c_{k}$ holds
$c_{k}(a_{u},c_{uv},b_{v})=0$ for some $c_{uv}\in F(uv)$, $a_{u}\in F(u)$ and
$b_{v}\in F(v)$. $\square$

Some relations between intersected-networks defined in Definition 39 are as
follows:

(i) $N_{int}(t-1)\subset N_{int}(t)$, $\Lambda(t-1)\subset\Lambda(t)$ and
$\mathcal{E}(t-1)\subset\mathcal{E}(t)$.

(ii) $N_{int}(t)=[O_{j_{1}},O_{j_{2}},\dots,O_{j_{m_{i}}}]\langle
N_{int}(t-1),N_{int}(t-2),\dots,N_{int}(t-k)\rangle$ for operations
$O_{j_{i}}\in\textbf{{O}}$ with $k\geq 1$, where
$\textbf{{O}}=\\{O_{k}\\}^{n}_{k=1}$ is a _network operation base_.

Moreover, we are interesting that each hypernetwork $N_{int}(t)$ obeys some
topological structure properties, such as self-similarity, scale-free
distribution, coefficient, small wold, homogeneousness, average, develop
velocity _etc_.

####  5.6.2 Scale-free intersected-networks

In 1999, Barabasi and Albert in [2] have shown that a _scale-free network_
$N(t)$ has its own _degree distribution_

$P(k)=\textrm{Pr}(x=k)\sim k^{-\lambda},~{}2<\lambda<3$ (59)

where $P(k)$ is the _power-law distribution_ of a vertex joined with $k$
vertices in $N(t)$.

Since an intersected-network $N_{int}(t)$ admits a set-coloring
$F:V(N_{int}(t))\rightarrow\mathcal{E}(t)$, where $\mathcal{E}(t)$ is the
hyperedge set of a hypernetwork
$\mathcal{N}_{yper}(t)=(\Lambda(t),\mathcal{E}(t))$ at time step $t$. If
$N_{int}(t)$ is _scale-free_ , then a vertex $w\in V(N_{int}(t))$ joined with
$k$ vertices in $N_{int}(t)$ corresponds a hyperedge $e\in\mathcal{E}(t)$
holding $F(w)=e$, which means that the _hyperedge-degree distribution_ of the
hypernetwork $\mathcal{N}_{yper}(t)$ obeys the _power-law distribution_
defined in Eq.(59) too, thus, we say $\mathcal{N}_{yper}(t)$ to be a _scale-
free hypernetwork_.

According to Definition 39 and some fundamental characteristics of a _scale-
free network model_ summarized in [22], a _scale-free intersected-network_
$N_{int}(t)$ having $v_{net}(t)$ vertices and $e_{net}(t)$ edges holds the
following topological properties:

Fc-1. [2] Two _common mechanisms_. _Growth_ is $v_{net}(t)>v_{net}(t-1)$ and
$e_{net}(t)>e_{net}(t-1)$, and _Preferential attachment_ is defined by
$\Pi_{i}(t)=k_{i}(t)/\sum_{j}k_{j}(t)$, where $k_{i}(t)$ and $k_{j}(t)$ are
hyperedge-degrees of hyperedges $e_{i}$ and $e_{j}$ of the hyperedge set
$\mathcal{E}(t)$.

Fc-2. [2] There is a _dynamic equation_

$\frac{\partial k_{i}(t)}{\partial t}=m\Pi_{i}(t)$ (60)

Using the initial condition $k_{i}(t_{i})=m$ solves the hyperedge-degree
function $k_{i}(t)$ from the dynamic equation Eq.(60).

Fc-3. A _sum_ $\sum n_{i}(k_{i}(t))\frac{\partial k_{i}(t)}{\partial t}$,
where $n_{i}(k_{i}(t))$ is the number of hyperedges having hyperedge-degree
$k_{i}(t)$ in $N_{int}(t)$.

Fc-4. [2] A _hyperedge-degree distribution_ $P(k)\sim k^{-\gamma}$ and a
_hyperedge cumulative distribution_ $P_{cum}(k)\sim k^{1-\gamma}$ with
$2<\gamma<3$ hold

$P(k)=\frac{\partial P(k_{i}(t)<k)}{\partial k},\quad P(k)=-\frac{\partial
P_{cum}(k)}{\partial k}$ (61)

The hyperedge cumulative distribution is defined as

$P_{cum}(k)=\sum_{k\,^{\prime}\geq
k}\frac{|V(k\,^{\prime},t)|}{v_{net}(t)}\sim k^{1-\lambda},$ (62)

with $2<\lambda<3$, where $|V(k\,^{\prime},t)|$ is the number of hyperedges of
hyperedge-degree $k\,^{\prime}$ (Ref. [13]).

The _hyperedge hyperedge-cumulative distribution_ is

$P_{ecum}(k)=\sum_{k\,^{\prime}\geq k}\frac{E(k\,^{\prime},t)}{e_{net}(t)}\sim
k^{1-\delta}$ (63)

with $2<\delta<3$, where $E(k\,^{\prime},t)$ is the number of hyperedges
adjacent with hyperedges of hyperedge-degree $k\,^{\prime}$ greater than $k$
at time step $t$ (Ref. [21]), and the _$d$ -hyperedge-cumulative distribution_
is

$P^{deg}_{ecum}(k)=\sum_{k\,^{\prime}\geq
k}\frac{k\,^{\prime}E(k\,^{\prime},t)}{e_{net}(t)}\sim k^{1-\varepsilon}$ (64)

with $2<\varepsilon<3$ (Ref. [23]).

Fc-5. We define the _velocity_ $V_{elo}(N_{int}(t))$ of an intersected-network
$N_{int}(t)$ as

$V_{elo}(N_{int}(t))=\sqrt{\left[\frac{\partial v_{net}(t)}{\partial
t}\right]^{2}+\left[\frac{\partial e_{net}(t)}{\partial t}\right]^{2}}$ (65)

where $\frac{\partial v_{net}(t)}{\partial t}$ is the _v-velocity_ and
$\frac{\partial e_{net}(t)}{\partial t}$ is the _e-velocity_. The _speed
ratio_ of the intersected-network $N_{int}(t)$ is defined by $\frac{\partial
e_{net}(t)}{\partial t}\div\frac{\partial v_{net}(t)}{\partial t}$, so the
_average hyperedge-degree_ $\langle k\rangle(t)$ of the hyperedge set
$\mathcal{E}(t)$ holds

$\frac{\partial e_{net}(t)}{\partial t}\sim\langle k\rangle(t)\cdot
b\cdot\frac{\partial v_{net}(t)}{\partial t}$ (66)

where $b$ is a constant (Ref. [23]).

Fc-6. If the scale-free hypernetwork $N_{int}(t)$ is _sparse_ (Ref. [14]), so
its average hyperedge-degree $\langle k\rangle(t)$ is approximate to a
constant, or

$e_{net}(t)\sim\textrm{O}(v_{net}(t)\ln[v_{net}(t)])$ (67)

DGN-intersected-network. Suppose that an intersected-network $N_{int}(t)$ is a
_deterministic growing intersected-network_ having $v_{net}(t)$ vertices and
$e_{net}(t)$ edges at time step $t$, we call $N_{int}(t)$ a _DGN-intersected-
network_.

The authors [22] have considered that a DGN-intersected-network $N_{int}(t)$
satisfies a system of linear equations (_linear growth_)

$v_{net}(t)=a_{v}t+b_{v},\quad e_{net}(t)=a_{e}t+b_{e}$ (68)

with $a_{v}>0$ and $a_{e}>0$. As known, Barabasi-Albert’s model holds Eq.(68)
true (Ref. [2]). Very often, a system of non-linear equations (_exponential
growth_) appeared in many literature is

$v_{net}(t)=a_{v}r\,^{t}+b_{v},\quad e_{net}(t)=a_{e}s\,^{t}+b_{e}$ (69)

with $a_{v}>0$, $a_{e}>0$, $|r|\neq 1$ and $|s|\neq 1$.

There are deterministic growing network models holding Eq.(69), such as the
Sierpinski model $S(t)$ with $r=3$ (Ref. [18]), the Recursive tree model
$R(t)$ with $r=q+1$ for $q\geq 2$ (Ref. [15]), and the Apollonian model $A(t)$
with $r=m(d+1)$ (Ref. [19]), we call them as _$r$ -rank models_, a tripartite
model introduced in [16] and a 4-partite model appeared in [20] both are
_2-rank models_.

####  5.6.3 Intersected-network lattices

Let $\textbf{{O}}=\\{O_{1},O_{2},\dots,O_{n}\\}$ be the set of graph
operations [36]. So, we call a subset $\textbf{{O}}_{i}=\\{O_{i,1}$,
$O_{i,2}$, $\dots,O_{i,m_{i}}\\}\subset\textbf{{O}}$ a _$W$ -operation base_
if each graph operation $O_{i,j}$ is not a compound of others, and obeys the
_$W$ -property_. We use symbol $a_{k}O_{i,k}$ to indicate $a_{k}$ copies of
$O_{i,k}\in\textbf{{O}}_{i}$ for $k\in[1,m_{i}]$, and
$O_{j_{1}},O_{j_{2}},\dots,O_{j_{B}}$ is a permutation of these graph
operations $a_{1}O_{i,1}$, $a_{2}O_{i,2}$, $\dots$, $a_{m_{i}}O_{i,m_{i}}$,
where $B=\sum^{m_{i}}_{k=1}a_{k}$.

We implement these graph operations $O_{j_{1}},O_{j_{2}},\dots,O_{j_{B}}$ to
an intersected-network $N_{int}(t)$, one by one, in the following way: The
first graph operation $O_{j_{1}}$ is implemented to an intersected-network
$N_{int}(t)$, the resultant intersected-network is denoted as
$G_{1}=O_{j_{1}}(N_{int}(t))$, and we have $G_{2}=O_{j_{2}}(G_{1})$,
$G_{3}=O_{j_{3}}(G_{2})$, $\dots$, $G_{B}=O_{j_{B}}(G_{B-1})$. We write the
last intersected-network $G_{B}=N_{int}(t)\lhd|^{m_{i}}_{k=1}\langle
a_{k}O_{i,k}\rangle$ for integrity. Simulating with the lattices introduced in
[36], we define an _operation intersected-network lattice_ as follows

${\begin{split}\textbf{{L}}\big{(}\textbf{{F}}(t)\lhd
Z^{0}\textbf{{O}}_{i}\big{)}=&\big{\\{}N_{int}(t)\lhd|^{m_{i}}_{k=1}\langle
a_{k}O_{i,k}\rangle:~{}a_{k}\in
Z^{0},O_{i,k}\in\textbf{{O}}_{i},N_{int}(t)\in\textbf{{F}}(t)\big{\\}}\end{split}}$
(70)

with $\sum^{m_{i}}_{k=1}a_{k}\geq 1$, where $\textbf{{F}}(t)$ is the set of
intersected-networks, and $\textbf{{O}}_{i}$ is called _$W$ -operation base_.

A _$W$ -operation base_ $\textbf{{O}}_{i}$ is called a _scale-free operation
base_ if each operation $O_{i,k}\in\textbf{{O}}_{i}$ is a _scale-free
operation_ defined by $P(k)\sim k^{-\lambda_{i,k}}$ with $\lambda_{i,k}>2$. We
call the following set

${\begin{split}&\textbf{{L}}\big{(}\textbf{{F}}_{scale}(t)\lhd
Z^{0}\textbf{{O}}_{i}\big{)}=\big{\\{}N_{int}(t)\lhd|^{m_{i}}_{k=1}\langle
b_{k}O_{i,k}\rangle:~{}b_{k}\in
Z^{0},O_{i,k}\in\textbf{{O}}_{i},N_{int}(t)\in\textbf{{F}}_{scale}(t)\big{\\}}\end{split}}$
(71)

_scale-free operation intersected-network lattice_ with
$\sum^{m_{i}}_{k=1}b_{k}\geq 1$, where $\textbf{{F}}_{scale}(t)$ is the set of
scale-free intersected-networks $N_{int}(t)$, and $\textbf{{O}}_{i}$ is a
_scale-free operation base_ such that each network of
$\textbf{{L}}\big{(}\textbf{{F}}_{scale}(t)\lhd Z^{0}\textbf{{O}}_{i}\big{)}$
is scale-free, in other word, $\textbf{{L}}\big{(}\textbf{{F}}_{scale}(t)\lhd
Z^{0}\textbf{{O}}_{i}\big{)}$ is really a _scale-free network generator_.

##  6 Conclusion

We think: A relation between the elements of a hyperedge set is also called an
operation, and there are not only common elements between the two elements of
a hyperedge set, but also other characteristics in common, we have shown our
ideas in Definition 27. Main works, in this article, are in the following:

  1. $\ast$

A set-colored graph $G$ is a topological expression of a hyperedge set
$\mathcal{E}$ of subsets of a finite set $\Lambda$, also, a visualization of
hypergraphs, however, the visualization methods are varied.

  2. $\ast$

Any connected graph is the intersected-graph of a hypergraph, and designed
colored-graphic lattices that are closed to some $W$-type coloring, that is,
each graph of this lattice admits this $W$-type coloring.

  3. $\ast$

By means of intersected-graphs, we have studied some characters of
hypergraphs, such as hyperedge connectivity, hyperedge coloring, hyper-total
coloring, set-set-colorings, compound hypergraphs, and so on.

  4. $\ast$

Several feasible and efficient algorithms for producing set-colorings of
graphs are distributed here.

  5. $\ast$

On the other hand, we have explored _hypergraph homomorphism_ shown in Eq.(55)
and _hypergraph lattices_ shown in Eq.(37) and Eq.(40).

  6. $\ast$

We try to apply hypergraph and set-coloring to various homomorphic
encryptions, and have introduced the hyperedge-splitting operation and the
hyperedge-coinciding operation of hypergraphs. Clearly, it is important to
investigate operations on hypergraphs, and it will help us to dig deeply the
property of hypergraphs and build up efficient tools for applying hypergraphs
in real world.

We have applied set-colorings and intersected-graphs to observe indirectly and
characterize properties of hypergraphs, no doubt, there are other methods to
observe and study hypergraphs, and discover more properties of hypergraphs.
We, also, hope that intersected-graphs mentioned here can be applied to real
situations, such as network security, privacy protection, as well as anti-
quantum computing in asymmetric cryptography. Predictably, hypergraph theory
will be important application in the future resisting AI attacks equipped
quantum computer, although the research achievements and application reports
of hypergraphs are far less than that of popular graphs.

Metaverse will be an important scene for the theoretical and application
research of hypergraphs and hypernetworks. It can also be said that the
development of Metaverse will greatly promote the theoretical and application
research of hypergraphs and hypernetworks.

Graphs are codes, and codes are graphs. There’s endless stuff to explore, it’s
interesting and beautiful, and a source of a lot of great questions [10].

## Acknowledgment

The author, _Bing Yao_ , was supported by the National Natural Science
Foundation of China under grants No. 61163054, No. 61363060 and No. 61662066.

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