Introduction for SuperHyperGraph Labeling and MultiLabeling

1. Preliminaries

This section fixes the terminology and notation used throughout the paper. Unless stated otherwise, every graph considered here is finite.

1.1. Labeling Graph

Graph theory investigates mathematical models built from vertices and edges that capture pairwise relations, network structure, and connectivity [1,2]. Graph labeling assigns numbers or symbols to vertices and/or edges of a graph under rules, modeling constraints, optimization, or communication [3,4,5,6].

Definition 1

(Graph labeling). [3,4,5,6] Let $G=(V,E)$ be a finite (simple) graph. A graph labeling is a choice of label sets $L_V$ (for vertices) and $L_E$ (for edges), together with functions

$${\ell }_V:V\to L_V\mathit{and}/\mathit{or}{\ell }_E:E\to L_E.$$

A labeling may be required to satisfy additional constraints, depending on the context. Typical instances include:

• Proper vertex-coloring: $L_V=\{1,\dots ,k\}$ and for every edge $uv\in E$, ${\ell }_V\left(u\right)\ne {\ell }_V\left(v\right)$.
• $L(p,q)$-labeling: $L_V\subseteq \mathbb{Z}$ such that for adjacent $u,v$ one has $|{\ell }_V\left(u\right)-{\ell }_V\left(v\right)|\ge p$, and for vertices at distance 2 one has $|{\ell }_V\left(u\right)-{\ell }_V\left(v\right)|\ge q$.
• Edge labelings(e.g., graceful, harmonious): constraints are imposed on ${\ell }_E$ (and sometimes on ${\ell }_V$) to control the multiset of induced values.

When only one of ${\ell }_V$ or ${\ell }_E$ is present, we speak of avertex labelingor anedge labeling, respectively.

Example 1

($L(2,1)$-labeling of a path). Consider the path $P_5$ with vertices $v_1,v_2,v_3,v_4,v_5$ and edges $v_i{v}_{i+1}$ ($i=1,\dots ,4$). Define a vertex labeling ${\ell }_V:V\left(P_5\right)\to \mathbb{Z}$ by

$${\ell }_V\left(v_1\right)=0,{\ell }_V\left(v_2\right)=2,{\ell }_V\left(v_3\right)=4,{\ell }_V\left(v_4\right)=1,{\ell }_V\left(v_5\right)=3.$$

We verify the $L(2,1)$ constraints:

• For each edge $v_i{v}_{i+1}$, the label gap is at least 2:

  $$|{\ell }_V\left(v_1\right)-{\ell }_V\left(v_2\right)|=|0-2|=2,|{\ell }_V\left(v_2\right)-{\ell }_V\left(v_3\right)|=|2-4|=2,$$

  $$|{\ell }_V\left(v_3\right)-{\ell }_V\left(v_4\right)|=|4-1|=3,|{\ell }_V\left(v_4\right)-{\ell }_V\left(v_5\right)|=|1-3|=2.$$
• For distance-2 pairs, the gap is at least 1:

  $$|{\ell }_V\left(v_1\right)-{\ell }_V\left(v_3\right)|=|0-4|=4,$$

  $$|{\ell }_V\left(v_2\right)-{\ell }_V\left(v_4\right)|=|2-1|=1,$$

  $$|{\ell }_V\left(v_3\right)-{\ell }_V\left(v_5\right)|=|4-3|=1.$$

Hence ${\ell }_V$ is a valid $L(2,1)$-labeling of $P_5$.

1.2. SuperHyperGraphs

A finite hypergraph extends the classical notion by allowing hyperedges to join arbitrary nonempty subsets of the vertex set, thereby representing multiway interactions [7,8,9]. Pushing this idea further, a finite SuperHyperGraph arises by iterating the powerset construction, which yields nested families of vertex- and edge-sets and thus encodes multi-layer relationships [10,11,12,13,14,15]. Such models are useful in, for example, molecular design, complex-network analysis, and advanced signal-processing pipelines [16,17]. Unless stated otherwise, the index n in ${\mathcal{P}}_n(·)$ and in an n–SuperHyperGraph is taken to be nonnegative.

Definition 2

(Base set). A base set S is the ambient universe of discourse:

$$S=\{x\mid x\mathit{belongs}\mathit{to}\mathit{the}\mathit{context}\mathit{under}\mathit{consideration}\}.$$

Every object occurring in $\mathcal{P}\left(S\right)$ or in any iterated powerset ${\mathcal{P}}_n\left(S\right)$ is, by definition, a subset ultimately formed from elements of S.

Definition 3

(Powerset). (see [18,19,20]) For a set S, the powerset $\mathcal{P}\left(S\right)$ is the collection of all subsets of S:

$$\mathcal{P}\left(S\right)=\{A\subseteq S\}.$$

In particular, both the empty set ∅ and S itself lie in $\mathcal{P}\left(S\right)$.

Definition 4

(Hypergraph). [21,22] A hypergraph is an ordered pair $H=(V,E)$ with

• a finite vertex set V, and
• a finite family E of nonempty subsets of V, whose members are called hyperedges.

Hypergraphs naturally encode interactions involving more than two participants.

Example 2 (Hypergraph — project teams sharing resources (real life)). Let the employees be the vertex set

$$V=\{\mathrm{Alice},\mathrm{Bob},\mathrm{Chen},\mathrm{Dina}\}.$$

Define the family of hyperedges

$$E=\left\{\{\mathrm{Alice},\mathrm{Bob},\mathrm{Chen}\},\{\mathrm{Bob},\mathrm{Dina}\},\{\mathrm{Alice},\mathrm{Dina}\}\right\}.$$

Interpretation. Each hyperedge is ateamthat jointly uses a shared resource (e.g., a meeting room or a code repository). Thus $H=(V,E)$ is a finite hypergraph: it records not only pairwise collaborations ($\{\mathrm{Bob},\mathrm{Dina}\}$) but also a three-person collaboration ($\{\mathrm{Alice},\mathrm{Bob},\mathrm{Chen}\}$).

Definition 5

(n-th powerset). [23,24,25] For a set X, define ${\mathcal{P}}_1\left(X\right)=\mathcal{P}\left(X\right)$ and, for $n\ge 1$,

$${\mathcal{P}}_{n+1}\left(X\right)=\mathcal{P}\left({\mathcal{P}}_n\left(X\right)\right).$$

When excluding the empty set, write ${\mathcal{P}}_n^{*}\left(X\right)={\mathcal{P}}_n\left(X\right)\setminus \{\varnothing \}.$

Example 3

(n-th powerset — explicit small instance). Take $X=\{p,q\}$. Then

$${\mathcal{P}}_1\left(X\right)=\mathcal{P}\left(X\right)=\{⌀,\left\{p\right\},\left\{q\right\},\{p,q\}\}.$$

The second-level powerset is the powerset of this 4-element set, hence $|{\mathcal{P}}_2\left(X\right)|=2^4=16$, for example it contains

$$\left\{\left\{p\right\}\right\},\left\{\left\{q\right\}\right\},\{\left\{p\right\},\left\{q\right\}\},\{⌀,\left\{p\right\},\left\{q\right\},\{p,q\}\}={\mathcal{P}}_1\left(X\right).$$

If we exclude the empty set at each step, then

$${\mathcal{P}}_1^{*}\left(X\right)=\{\left\{p\right\},\left\{q\right\},\{p,q\}\},{\mathcal{P}}_2^{*}\left(X\right)={\mathcal{P}}_2\left(X\right)\setminus \{⌀\}.$$

This illustrates how iterating $\mathcal{P}(·)$ builds higher “layers’’ of set families.

Definition 6

(n–SuperHyperGraph). (see [26,27]) Let $V_0$ be a finite, nonempty base set and define

$${\mathcal{P}}^0\left(V_0\right)V_0,{\mathcal{P}}^{k+1}\left(V_0\right)\mathcal{P}\left({\mathcal{P}}^k\left(V_0\right)\right)(k\in \mathbb{N}).$$

For $n\ge 0$, an n–SuperHyperGraph on $V_0$ is a pair

$${\mathrm{SHG}}^{\left(n\right)}=(V,E)$$

with

$$V\subseteq {\mathcal{P}}^n\left(V_0\right)\mathrm{and}E\subseteq \mathcal{P}\left(V\right)\setminus \{\varnothing \}.$$

Members of V are the n–supervertices, while members of E are the n–superedges (each n–superedge is a nonempty subset of V).

Example 4

(n–SuperHyperGraph — families of task-sets (real life)). Let the base set of atomic tasks be $V_0=\{\mathrm{a},\mathrm{b},\mathrm{c}\}$. Then ${\mathcal{P}}^1\left(V_0\right)=\mathcal{P}\left(V_0\right)$ consists of all task-sets, and ${\mathcal{P}}^2\left(V_0\right)=\mathcal{P}\left(\mathcal{P}\left(V_0\right)\right)$ consists offamiliesof task-sets. Choose the $n=2$ supervertex set

$$V=\left\{F_1,F_2\right\},F_1=\left\{\left\{\mathrm{a}\right\},\left\{\mathrm{b}\right\}\right\},F_2=\left\{\left\{\mathrm{b}\right\},\left\{\mathrm{c}\right\}\right\}.$$

Define the superedge family

$$E=\left\{\left\{F_1\right\},\left\{F_2\right\},\{F_1,F_2\}\right\}\subseteq \mathcal{P}\left(V\right)\setminus \{⌀\}.$$

Interpretation. Each supervertex $F_i$ is aplan family (a set of admissible task-sets); a superedge groups one or several plan families that are considered together (e.g., combined scenarios). Hence ${\mathrm{SHG}}^{\left(2\right)}=(V,E)$ is a finite 2–SuperHyperGraph on $V_0$.

2. Main Results

This section presents the principal findings of the paper.

2.1. Hypergraph labeling

HyperGraph labeling assigns labels to vertices and hyperedges of hypergraphs, encoding multi-participant interactions, scheduling, resource distribution, or network optimization (cf.[28,29,30,31,32]).

Definition 7 (Primal (2-section) graph). Given a hypergraph $H=(V,E)$, its primal graph (also called the 2-section) is

$$G\left(H\right):=\left(V,E^{\prime }\right),E^{\prime }:=\left\{\{u,v\}\subseteq V|\exists e\in E\mathrm{with}\{u,v\}\subseteq e\right\}.$$

Distances between vertices of H are measured in $G\left(H\right)$ and denoted ${dist}_H$.

Definition 8 (Hypergraph labeling (schema-based)). Let $H=(V,E)$ be a hypergraph, and let $L_V,L_E$ be nonempty label sets (for vertices and hyperedges). A (vertex/edge) labeling of H is a pair of maps

$${\ell }_V:V\to L_V,{\ell }_E:E\to L_E,$$

where either map may be omitted if not used. Ahypergraph labeling schemais a first-order predicate $\mathsf{\Phi }(H,{\ell }_V,{\ell }_E)$ built from the incidence relation “$v\in e$”, the distance ${dist}_H$ on V (Definition 7), the equality/inequality on labels, and quantification over V and E. We say that $({\ell }_V,{\ell }_E)$ is a valid hypergraph labeling (for $\mathsf{\Phi }$)if $\mathsf{\Phi }(H,{\ell }_V,{\ell }_E)$ holds.

Remark 1

(Classical graph labelings as instances of $\mathsf{\Phi }$). Typical choices of Φ recover familiar graph labelings when H is 2-uniform:

• Proper vertex coloring:  $L_V=\{1,\dots ,k\}$ and $\mathsf{\Phi }\equiv (\forall \{u,v\}\in E){\ell }_V\left(u\right)\ne {\ell }_V\left(v\right)$.
• $L(p,q)$-labeling: $L_V\subseteq \mathbb{Z}$ and

  $$\mathsf{\Phi }\equiv \left(\forall u\ne v\in V\right)\left({dist}_H(u,v)=1\Rightarrow |{\ell }_V\left(u\right)-{\ell }_V\left(v\right)|\ge p\wedge {dist}_H(u,v)=2\Rightarrow |{\ell }_V\left(u\right)-{\ell }_V\left(v\right)|\ge q\right).$$
• Strong hypergraph coloring (a genuine hypergraph constraint): $L_V=\{1,\dots ,k\}$ and $\mathsf{\Phi }\equiv (\forall e\in E)\mathit{the}\mathit{labels}\{{\ell }_V\left(v\right)\mid v\in e\}\mathit{are}\mathit{pairwise}\mathit{distinct}$.

Example 5

(Strong hypergraph 3-coloring of a small hypergraph). Let $H=(V,E)$ with

$$V=\{1,2,3,4\},E=\left\{\{1,2,3\},\{3,4\}\right\}.$$

Take $L_V=\{r,g,b\}$ and let Φ be “strong coloring”: for every $e\in E$ the labels on e are pairwise distinct. Define

$${\ell }_V\left(1\right)=r,{\ell }_V\left(2\right)=g,{\ell }_V\left(3\right)=b,{\ell }_V\left(4\right)=r.$$

Verification.On $e_1=\{1,2,3\}$ we have $\{r,g,b\}$ (all distinct). On $e_2=\{3,4\}$ we have $\{b,r\}$ (distinct). Thus ${\ell }_V$ satisfies Φ and is a valid strong 3-coloring of H.

Example 6

($L(2,1)$-labeling lifted to a hypergraph via the 2-section). Let $H=(V,E)$ with

$$V=\{a,b,c,d,e\},E=\left\{\{a,b,c\},\{c,d\},\{d,e\}\right\}.$$

Its 2-section $G\left(H\right)$ has edges

$$\{a,b\},\{a,c\},\{b,c\},\{c,d\},\{d,e\}.$$

Let $L_V=\mathbb{Z}$ and let Φ be the hypergraph version of $L(2,1)$: for all distinct $u,v\in V$,

$${dist}_H(u,v)=1\Rightarrow |{\ell }_V\left(u\right)-{\ell }_V\left(v\right)|\ge 2,{dist}_H(u,v)=2\Rightarrow |{\ell }_V\left(u\right)-{\ell }_V\left(v\right)|\ge 1,$$

where ${dist}_H$ is computed in $G\left(H\right)$. Define labels

$${\ell }_V\left(a\right)=0,{\ell }_V\left(b\right)=2,{\ell }_V\left(c\right)=4,{\ell }_V\left(d\right)=1,{\ell }_V\left(e\right)=3.$$

Adjacency checks(distance 1 in $G\left(H\right)$):

$$|0-2|=2\left(\right\{a,b\left\}\right),|0-4|=4\left(\right\{a,c\left\}\right),|2-4|=2\left(\right\{b,c\left\}\right),|4-1|=3\left(\right\{c,d\left\}\right),|1-3|=2\left(\right\{d,e\left\}\right).$$

Distance-2 checks(one example per class):

$$a-d:a-c-d\Rightarrow |0-1|=1;b-d:b-c-d\Rightarrow |2-1|=1;$$

$$a-e:a-c-d-e\Rightarrow |0-3|=3;b-e:b-c-d-e\Rightarrow |2-3|=1.$$

All constraints hold, hence ${\ell }_V$ is a valid $L(2,1)$-labeling of the hypergraph H under Φ. When H is 2-uniform, this reduces to the classical $L(2,1)$-labeling of a graph.

Theorem 1

(Hypergraph labeling strictly generalizes graph labeling). Let $(G,\mathsf{\Psi })$ be any graph-labeling problem on a simple graph $G=(V,E_G)$, where $\mathsf{\Psi }(G,\lambda )$ is a predicate expressed using adjacency/distance in G and (in)equalities among labels. Regard G as the 2-uniform hypergraph $H=(V,E)$ with $E:=\{\{u,v\}\mid uv\in E_G\}$. Define the hypergraph labeling schema $\mathsf{\Phi }(H,{\ell }_V):=\mathsf{\Psi }\left(G\left(H\right),{\ell }_V\right)$. Then the following are equivalent:

$$\exists \lambda :V\to L\mathit{with}\mathsf{\Psi }(G,\lambda )⟺\exists {\ell }_V:V\to L\mathit{with}\mathsf{\Phi }(H,{\ell }_V).$$

Consequently, every graph labeling instance is an instance of hypergraph labeling; moreover, hypergraph-only constraints (e.g. strong hypergraph coloring) have no counterpart on graphs with only size-2 edges, so the inclusion is strict.

Proof. (⇒) Suppose $\lambda :V\to L$ satisfies $\mathsf{\Psi }(G,\lambda )$. Since $G\left(H\right)=G$ by construction of E from $E_G$, setting ${\ell }_V:=\lambda$ yields

$$\mathsf{\Phi }(H,{\ell }_V)\equiv \mathsf{\Psi }\left(G\left(H\right),{\ell }_V\right)=\mathsf{\Psi }(G,\lambda ),$$

so ${\ell }_V$ is a valid hypergraph labeling.

(⇐) Conversely, if ${\ell }_V:V\to L$ satisfies $\mathsf{\Phi }(H,{\ell }_V)$, then $\mathsf{\Psi }\left(G\left(H\right),{\ell }_V\right)$ holds. But $G\left(H\right)=G$, hence $\mathsf{\Psi }(G,{\ell }_V)$ holds, and $\lambda :={\ell }_V$ is a valid graph labeling.

Strictness follows because there exist valid $\mathsf{\Phi }$ that quantify over hyperedges of size $\ge 3$ (e.g. strong hypergraph coloring in Remark 1), which impose constraints on $\left|e\right|$-tuples of vertices inside a single hyperedge; such constraints cannot be expressed on a simple graph whose edges are only size-2 subsets without passing to cliques or auxiliary gadgets. □

2.2. SuperHyperGraph Labeling

SuperHyperGraph labeling assigns structured labels to vertices and superedges across nested powerset levels, capturing hierarchical multi-layered relationships and advanced constraints.

Definition 9 (SuperHyperGraph labeling (schema–based)). Let ${\mathsf{SHG}}^{\left(n\right)}=(V,E)$ and let $L_V,L_E$ be nonempty label sets for vertices and superedges. A (vertex/edge) labeling is a pair of maps

$${\ell }_V:V\to L_V,{\ell }_E:E\to L_E,$$

where either map may be omitted if not needed. A labeling schema is a first–order predicate

$$\mathsf{\Phi }\left({\mathsf{SHG}}^{\left(n\right)},{\ell }_V,{\ell }_E\right)$$

built from the incidence relation “$v\in e$” ($v\in V$, $e\in E$), equality/inequality on labels, the distance ${dist}_{\mathsf{SHG}}$ from the Definition, and (optionally) structural predicates on n–level objects (e.g. cardinalities, inclusion between members of $V\subseteq {\mathcal{P}}^n\left(V_0\right)$). We call $({\ell }_V,{\ell }_E)$ a valid SuperHyperGraph labeling (for Φ) if $\mathsf{\Phi }({\mathsf{SHG}}^{\left(n\right)},{\ell }_V,{\ell }_E)$ holds.

Remark 2

(Recovering classical schemas). By choosing Φ appropriately one recovers many standard labeling families:

• Proper vertex coloring: $L_V=\{1,\dots ,k\}$ and $(\forall \{u,v\}\in E^{\prime }){\ell }_V\left(u\right)\ne {\ell }_V\left(v\right)$, where $E^{\prime }$ is from the 2–section.
• $L(p,q)$–labeling: $L_V\subseteq \mathbb{Z}$ and, for all distinct $u,v\in V$,

  $${dist}_{\mathsf{SHG}}(u,v)=1\Rightarrow |{\ell }_V\left(u\right)-{\ell }_V\left(v\right)|\ge p,{dist}_{\mathsf{SHG}}(u,v)=2\Rightarrow |{\ell }_V\left(u\right)-{\ell }_V\left(v\right)|\ge q.$$
• Strong hypercoloring (genuinely hyper): $L_V=\{1,\dots ,k\}$ and $(\forall e\in E)$ the labels $\{{\ell }_V\left(v\right):v\in e\}$ are pairwise distinct.

Example 7

(Classical $L(2,1)$ as a SuperHyperGraph labeling (the case $n=0$)). Let $G=P_5$ be the path on vertices $v_1,\dots ,v_5$. Form the 0–SuperHyperGraph ${\mathsf{SHG}}^{\left(0\right)}=(V,E)$ with $V=\{v_1,\dots ,v_5\}$ and $E=\left\{\{v_i,v_{i+1}\}:i=1,2,3,4\right\}$. Let Φ be the $L(2,1)$ schema of Remark 2 with $L_V=\mathbb{Z}$ and distances taken in $G\left({\mathsf{SHG}}^{\left(0\right)}\right)=G$. Define

$${\ell }_V\left(v_1\right)=0,{\ell }_V\left(v_2\right)=2,{\ell }_V\left(v_3\right)=4,{\ell }_V\left(v_4\right)=1,{\ell }_V\left(v_5\right)=3.$$

Adjacency (distance 1) checks:$|0-2|=2$, $|2-4|=2$, $|4-1|=3$, $|1-3|=2$.Distance 2 checks:$|0-4|=4$, $|2-1|=1$, $|4-3|=1$. All constraints hold, so ${\ell }_V$ is a valid SuperHyperGraph $L(2,1)$ labeling. In this case we exactly recover the classical graph labeling.

Example 8

(An $n=1$ SuperHyperGraph with an $L(2,1)$–type labeling on overlapping sets). Let $V_0=\{a,b,c,d\}$ and consider the 1–level supervertices

$$A=\{a,b\},B=\{b,c\},C=\{c,d\},D=\{a,d\}\in \mathcal{P}\left(V_0\right).$$

Set $V=\{A,B,C,D\}$ and define superedges

$$E_1=\{A,B,C\},E_2=\{A,D\},E_3=\{C,D\}.$$

Thus ${\mathsf{SHG}}^{\left(1\right)}=(V,\{E_1,E_2,E_3\})$. Its 2–section $G\left({\mathsf{SHG}}^{\left(1\right)}\right)$ has edges

$$\{A,B\},\{A,C\},\{B,C\}\left(\mathrm{from}{E}_1\right),\{A,D\}\left(\mathrm{from}{E}_2\right),\{C,D\}\left(\mathrm{from}{E}_3\right).$$

Let $L_V=\mathbb{Z}$ and impose the $L(2,1)$ schema from Remark 2 with distances taken in $G\left({\mathsf{SHG}}^{\left(1\right)}\right)$. Define the labeling

$${\ell }_V\left(A\right)=0,{\ell }_V\left(B\right)=2,{\ell }_V\left(C\right)=5,{\ell }_V\left(D\right)=7.$$

Adjacency (distance 1) checks:

$$|0-2|=2\left(\right\{A,B\left\}\right),|0-5|=5\left(\right\{A,C\left\}\right),|2-5|=3\left(\right\{B,C\left\}\right),|0-7|=7\left(\right\{A,D\left\}\right),|5-7|=2\left(\right\{C,D\left\}\right).$$

Distance 2 check: B and D have distance 2 (via A or C), and $|2-7|=5\ge 1$. Hence ${\ell }_V$ satisfies the $L(2,1)$ constraints on this genuinely superhyper (level $n=1$) instance. Note that the vertices here aresets of base elements, and superedges may have size 3, a setting that goes beyond ordinary graphs.

Theorem 2

(SuperHyperGraph labeling generalizes graph and hypergraph labeling). Let $(G,\mathsf{\Psi })$ be any graph–labeling problem on a simple graph $G=(V,E_G)$, with predicate $\mathsf{\Psi }(G,\lambda )$ expressed in terms of adjacency/distance in G and (in)equalities among labels. Let $(H,\mathsf{\Psi })$ be any hypergraph–labeling problem on a hypergraph $H=(V,E_H)$ where distances are computed in the hypergraph 2–section.

Then there exist $n\in \left\{0\right\}$, a SuperHyperGraph ${\mathsf{SHG}}^{\left(n\right)}$, and a labeling schema Φ such that:

$$\exists \lambda :V\to L\mathrm{with}\mathsf{\Psi }(G,\lambda )⟺\exists {\ell }_V:V\to L\mathrm{with}\mathsf{\Phi }({\mathsf{SHG}}^{\left(0\right)},{\ell }_V),$$

and

$$\exists \lambda :V\to L\mathrm{with}\mathsf{\Psi }(H,\lambda )⟺\exists {\ell }_V:V\to L\mathrm{with}\mathsf{\Phi }({\mathsf{SHG}}^{\left(0\right)},{\ell }_V).$$

Consequently, SuperHyperGraph labeling strictly contains graph labeling (the case of 2–uniform hyperedges) and hypergraph labeling (the case $n=0$ with general hyperedges).

Proof. 

For the graph case, let $n=0$, $V_0:=V$, and define ${\mathsf{SHG}}^{\left(0\right)}=(V,E)$ with $E:=\{\{u,v\}\subseteq V\mid uv\in E_G\}$. Then $G\left({\mathsf{SHG}}^{\left(0\right)}\right)=G$ by construction. Define $\mathsf{\Phi }\left({\mathsf{SHG}}^{\left(0\right)},{\ell }_V\right):=\mathsf{\Psi }\left(G\left({\mathsf{SHG}}^{\left(0\right)}\right),{\ell }_V\right)$. Hence $\mathsf{\Psi }(G,\lambda )$ holds iff $\mathsf{\Phi }({\mathsf{SHG}}^{\left(0\right)},\lambda )$ holds, giving the first equivalence.

For the hypergraph case, again take $n=0$, $V_0:=V$, and set ${\mathsf{SHG}}^{\left(0\right)}=(V,E_H)$. By definition of the hypergraph 2–section, $G\left({\mathsf{SHG}}^{\left(0\right)}\right)$ is exactly the primal graph used to measure distances in $\mathsf{\Psi }(H,·)$. Put $\mathsf{\Phi }\left({\mathsf{SHG}}^{\left(0\right)},{\ell }_V\right):=\mathsf{\Psi }\left(H,{\ell }_V\right)$, interpreting all distance/adjacency relations through $G\left({\mathsf{SHG}}^{\left(0\right)}\right)$. The same identity–of–structures argument yields the second equivalence.

Strict containment follows since for $n\ge 1$ one can add constraints that speak about the internal structure of n–level supervertices (e.g. intersection/nonintersection of members when $V\subseteq \mathcal{P}\left(V_0\right)$), which cannot be expressed on ordinary graphs nor on hypergraphs with $n=0$ without expanding the vertex set. □

2.3. Graph MultiLabeling

Graph MultiLabeling assigns multiple simultaneous labels to vertices and edges, supporting layered constraints, diverse applications, and richer graph optimization models.

Definition 10

(Graph MultiLabeling). Fix nonnegative integers $p,q$. For a graph $G=(V,E)$, choose nonemptyvertex-label alphabets$L_V^{\left(1\right)},\dots ,L_V^{\left(p\right)}$ and nonemptyedge-label alphabets$L_E^{\left(1\right)},\dots ,L_E^{\left(q\right)}$. AGraph MultiLabelingon G is the tuple of maps

$${\ell }_V=\left({\ell }_V^{\left(1\right)},\dots ,{\ell }_V^{\left(p\right)}\right),{\ell }_V^{\left(a\right)}:V\to L_V^{\left(a\right)}(1\le a\le p),$$

$${\ell }_E=\left({\ell }_E^{\left(1\right)},\dots ,{\ell }_E^{\left(q\right)}\right),{\ell }_E^{\left(b\right)}:E\to L_E^{\left(b\right)}(1\le b\le q).$$

Equivalently, ${\ell }_V:V\to {\prod }_{a=1}^p{L}_V^{\left(a\right)}$, ${\ell }_E:E\to {\prod }_{b=1}^q{L}_E^{\left(b\right)}$ with ${\ell }_V\left(v\right)=\left({\ell }_V^{\left(1\right)}\left(v\right),\dots ,{\ell }_V^{\left(p\right)}\left(v\right)\right)$ and similarly for edges.

A MultiLabeling schema is a first-order predicate

$$\mathsf{\Phi }\left(G;{\ell }_V,{\ell }_E\right)$$

built from adjacency/distances in G, the incidence relation $u\in e$, the label components ${\ell }_V^{\left(a\right)}\left(u\right)$, ${\ell }_E^{\left(b\right)}\left(e\right)$, and fixed relations on these alphabets (e.g. equality, order, arithmetic, or application-specific constraints). We say that $({\ell }_V,{\ell }_E)$ is a valid Graph MultiLabeling for $\mathsf{\Phi }$if $\mathsf{\Phi }\left(G;{\ell }_V,{\ell }_E\right)$ holds.

Remark 3

(Typical coordinatewise constraints). Many familiar labeling families appear as single coordinates:

• Proper coloring on coordinate a: $L_V^{\left(a\right)}=\{1,\dots ,k\}$ and $(\forall \{u,v\}\in E){\ell }_V^{\left(a\right)}\left(u\right)\ne {\ell }_V^{\left(a\right)}\left(v\right)$.
• $L(p,q)$ on coordinate a: $L_V^{\left(a\right)}\subseteq \mathbb{Z}$ with

  $${dist}_G(u,v)=1\Rightarrow |{\ell }_V^{\left(a\right)}\left(u\right)-{\ell }_V^{\left(a\right)}\left(v\right)|\ge p,{dist}_G(u,v)=2\Rightarrow |{\ell }_V^{\left(a\right)}\left(u\right)-{\ell }_V^{\left(a\right)}\left(v\right)|\ge q.$$
• Edge capacities on coordinate b: $L_E^{\left(b\right)}=\{1,\dots ,C\}$ with cross-constraints such as ${\ell }_E^{\left(b\right)}\left(\{u,v\}\right)\ge f\left({\ell }_V^{\left(a\right)}\left(u\right),{\ell }_V^{\left(a\right)}\left(v\right)\right)$ for a fixed function f.

Coordinates may also becoupled, e.g. requiring that a time-slot label and a color label jointly avoid conflicts.

Example 9

(Two-coordinate vertex MultiLabeling on a path: coloring +$L(2,1)$). Let $G=P_5$ with vertices $v_1,\dots ,v_5$ and edges $v_i{v}_{i+1}$ ($i=1,\dots ,4$). Choose $p=2$, $q=0$ with

$$L_V^{\left(1\right)}=\{r,g,b\}\left(\mathrm{colors}\right),L_V^{\left(2\right)}=\mathbb{Z}\left(\mathit{integers}\right).$$

Define ${\ell }_V=({\ell }_V^{\left(1\right)},{\ell }_V^{\left(2\right)})$ by

$${\ell }_V^{\left(1\right)}:(v_1,\dots ,v_5)\mapsto (r,g,b,g,r),$$

$${\ell }_V^{\left(2\right)}:(v_1,\dots ,v_5)\mapsto (0,2,4,1,3).$$

Schema Φ requires simultaneously:

(C)

Proper coloringon coordinate 1: for each edge $v_i{v}_{i+1}$, ${\ell }_V^{\left(1\right)}\left(v_i\right)\ne {\ell }_V^{\left(1\right)}\left(v_{i+1}\right)$.

(N)

$L(2,1)$on coordinate 2: for all distinct $u,v$,

$${dist}_G(u,v)=1\Rightarrow |{\ell }_V^{\left(2\right)}\left(u\right)-{\ell }_V^{\left(2\right)}\left(v\right)|\ge 2,{dist}_G(u,v)=2\Rightarrow |{\ell }_V^{\left(2\right)}\left(u\right)-{\ell }_V^{\left(2\right)}\left(v\right)|\ge 1.$$

Verification. (C) Adjacent pairs are $(r,g)$, $(g,b)$, $(b,g)$, $(g,r)$, all unequal. (N) Adjacent gaps: $|0-2|=2$, $|2-4|=2$, $|4-1|=3$, $|1-3|=2$. Distance-2 gaps: $|0-4|=4$, $|2-1|=1$, $|4-3|=1$. Hence $({\ell }_V,{\ell }_E=⌀)$ satisfies Φ. This is a genuinemulti-labeling: two coordinated vertex labelings enforced at once.

Example 10

(Vertex & edge MultiLabeling with cross-constraints). Let G be the 4-cycle $C_4$ on $v_1,v_2,v_3,v_4$ (in order). Take $p=1$, $q=1$ with $L_V^{\left(1\right)}=\{A,B\}$ (two roles) and $L_E^{\left(1\right)}=\{1,2\}$ (capacity classes). Define

$${\ell }_V^{\left(1\right)}\left(v_1\right)=A,{\ell }_V^{\left(1\right)}\left(v_2\right)=B,{\ell }_V^{\left(1\right)}\left(v_3\right)=A,{\ell }_V^{\left(1\right)}\left(v_4\right)=B,$$

$${\ell }_E^{\left(1\right)}\left(\{v_1,v_2\}\right)=2,{\ell }_E^{\left(1\right)}\left(\{v_2,v_3\}\right)=1,{\ell }_E^{\left(1\right)}\left(\{v_3,v_4\}\right)=2,{\ell }_E^{\left(1\right)}\left(\{v_4,v_1\}\right)=1.$$

Schema Φ requires:

• Adjacent vertices must have different roles: ${\ell }_V^{\left(1\right)}\left(u\right)\ne {\ell }_V^{\left(1\right)}\left(v\right)$ for all $\{u,v\}\in E$ (a 2-coloring).
• Edge capacity must dominate therole disparity:

  $${\ell }_E^{\left(1\right)}\left(\{u,v\}\right)\ge \left\{\begin{array}{cc}2,\hfill & \mathrm{if}\{{\ell }_V^{\left(1\right)}\left(u\right),{\ell }_V^{\left(1\right)}\left(v\right)\}=\{A,B\},\hfill \\ 1,\hfill & \mathrm{if}{\ell }_V^{\left(1\right)}\left(u\right)={\ell }_V^{\left(1\right)}\left(v\right).\hfill \end{array}\right.$$

Verification.The vertex roles alternate $A,B,A,B$, so the first condition holds. Each edge has endpoints of different roles, and is labeled capacity 2 or 1 as above; precisely those with alternating endpoints have capacity 2, satisfying the inequality. Thus $({\ell }_V,{\ell }_E)$ is a valid MultiLabeling with coupled vertex/edge constraints.

Theorem 3

(Graph MultiLabeling generalizes classical graph labeling). Let $(G,\mathsf{\Psi })$ be any classical graph-labeling problem on $G=(V,E)$: i.e., choose a single label set L and a predicate $\mathsf{\Psi }(G,\lambda )$ over mappings $\lambda :V\to L$ (or $\lambda :E\to L$) that is expressed using graph structure and relations on L. Then there exists a MultiLabeling schema Φ with $p=1$ (and $q=0$ for a vertex-labeling, or $q=1$, $p=0$ for an edge-labeling) such that

$$\exists \lambda \mathrm{with}\mathsf{\Psi }(G,\lambda )⟺\exists ({\ell }_V,{\ell }_E)\mathrm{with}\mathsf{\Phi }\left(G;{\ell }_V,{\ell }_E\right).$$

Proof. 

Vertex-labeling case. Take $p=1$, $q=0$, and set $L_V^{\left(1\right)}:=L$. Define $\mathsf{\Phi }\left(G;\left({\ell }_V^{\left(1\right)}\right),⌀\right):=\mathsf{\Psi }\left(G,{\ell }_V^{\left(1\right)}\right)$. Then $\lambda$ satisfies $\mathsf{\Psi }$ iff ${\ell }_V^{\left(1\right)}:=\lambda$ satisfies $\mathsf{\Phi }$; existence is equivalent.

Edge-labeling case is identical with $p=0$, $q=1$, $L_E^{\left(1\right)}:=L$, and $\mathsf{\Phi }\left(G;⌀,\left({\ell }_E^{\left(1\right)}\right)\right):=\mathsf{\Psi }\left(G,{\ell }_E^{\left(1\right)}\right)$. □

2.4. HyperGraph MultiLabeling

HyperGraph MultiLabeling provides multiple coordinated labels to vertices and hyperedges, generalizing graph multilabeling and enabling multi-role representations of complex relationships.

Definition 11 (Primal (2–section) of a hypergraph). Given $H=(V,E)$, its primal graph (also 2–section) is

$$G\left(H\right):=(V,E^{\prime })\mathrm{with}{E}^{\prime }:=\left\{\{u,v\}\subseteq V\mid \exists e\in E:\{u,v\}\subseteq e\right\}.$$

We write ${\mathrm{dist}}_H(u,v)$ for the usual shortest–path distance of $u,v\in V$ taken in $G\left(H\right)$.

Definition 12

(HyperGraph MultiLabeling). Fix nonnegative integers $p,q$. Let $H=(V,E)$ be a hypergraph. Choose nonemptyvertex label alphabets$L_V^{\left(1\right)},\dots ,L_V^{\left(p\right)}$ and nonempty hyperedge label alphabets$L_E^{\left(1\right)},\dots ,L_E^{\left(q\right)}$. A HyperGraph MultiLabeling on H consists of the coordinate maps

$${\ell }_V=\left({\ell }_V^{\left(1\right)},\dots ,{\ell }_V^{\left(p\right)}\right),{\ell }_V^{\left(a\right)}:V\to L_V^{\left(a\right)}(1\le a\le p),$$

$${\ell }_E=\left({\ell }_E^{\left(1\right)},\dots ,{\ell }_E^{\left(q\right)}\right),{\ell }_E^{\left(b\right)}:E\to L_E^{\left(b\right)}(1\le b\le q).$$

Equivalently, a single vertex map ${\ell }_V:V\to {\prod }_{a=1}^p{L}_V^{\left(a\right)}$ together with a single edge map ${\ell }_E:E\to {\prod }_{b=1}^q{L}_E^{\left(b\right)}$.

A MultiLabeling schema is a first–order predicate

$$\mathsf{\Phi }\left(H;{\ell }_V,{\ell }_E\right)$$

built from the incidence relation $v\in e$, the distance ${\mathrm{dist}}_H$ on V, (and possibly fixed relations/operations on the alphabets, such as =, ≠, order, arithmetic, etc.). We say $({\ell }_V,{\ell }_E)$ is a valid HyperGraph MultiLabeling (for $\mathsf{\Phi }$)if $\mathsf{\Phi }\left(H;{\ell }_V,{\ell }_E\right)$ holds.

Remark 4

(Typical coordinatewise and cross–coordinate constraints). The schema Φ can encode, e.g.:

• Strong hyperedge coloring  on a vertex coordinate a: for every $e\in E$, the set $\{{\ell }_V^{\left(a\right)}\left(v\right):v\in e\}$ is pairwise distinct.
• $L(p,q)$–type spacing  on a vertex coordinate a: if ${\mathrm{dist}}_H(u,v)=1$ then $|{\ell }_V^{\left(a\right)}\left(u\right)-{\ell }_V^{\left(a\right)}\left(v\right)|\ge p$, and if ${\mathrm{dist}}_H(u,v)=2$ then $|{\ell }_V^{\left(a\right)}\left(u\right)-{\ell }_V^{\left(a\right)}\left(v\right)|\ge q$.
• Vertex–edge coupling: for each $e\in E$, a constraint linking ${\ell }_E^{\left(b\right)}\left(e\right)$ to an aggregate of $\{{\ell }_V^{\left(a\right)}\left(v\right):v\in e\}$ (sum, max, cardinality, etc.).

Example 11

(Two–coordinate vertex & one–coordinate edge MultiLabeling on a 3–uniform hypergraph). Let $V=\{a,b,c,d\}$ and $E=\{e_1,e_2\}$ with $e_1=\{a,b,c\}$ and $e_2=\{b,c,d\}$. Choose $p=2$, $q=1$ with

$$L_V^{\left(1\right)}=\{R,G,B\}\left(\mathit{colors}\right),L_V^{\left(2\right)}={\mathbb{Z}}_{\ge 0}\left(\mathit{workload}\right),$$

$$L_E^{\left(1\right)}={\mathbb{Z}}_{\ge 0}\left(\mathit{edge}\mathit{deadline}\right).$$

Define the labeling:

$${\ell }_V^{\left(1\right)}\left(a\right)=R,{\ell }_V^{\left(1\right)}\left(b\right)=G,{\ell }_V^{\left(1\right)}\left(c\right)=B,{\ell }_V^{\left(1\right)}\left(d\right)=G,$$

$${\ell }_V^{\left(2\right)}\left(a\right)=1,{\ell }_V^{\left(2\right)}\left(b\right)=2,{\ell }_V^{\left(2\right)}\left(c\right)=1,{\ell }_V^{\left(2\right)}\left(d\right)=3,$$

$${\ell }_E^{\left(1\right)}\left(e_1\right)=5,{\ell }_E^{\left(1\right)}\left(e_2\right)=7.$$

Schema Φ requires simultaneously:

(S) 

Strong hyperedge coloringon coordinate 1: within each $e\in E$, the colors are pairwise distinct.

(C) 

Capacity coupling: for each $e\in E$,

$$\sum _{v\in e}{\ell }_V^{\left(2\right)}\left(v\right)\le {\ell }_E^{\left(1\right)}\left(e\right).$$

Verification. (S) In $e_1=\{a,b,c\}$ we have $(R,G,B)$ distinct; in $e_2=\{b,c,d\}$ we have $(G,B,G)$ not all distinct. Thus the strong constraint would fail on $e_2$. Instead, choose a weak variant for $e_2$: “at least two colors in each hyperedge’’; then $e_2$ uses $\{G,B\}$ and passes. (C) $e_1$: $1+2+1=4\le 5$; $e_2$: $2+1+3=6\le 7$. Hence with weak hyperedge coloring the tuple $({\ell }_V,{\ell }_E)$ satisfies Φ.

Example 12 (Distance–aware MultiLabeling (an $L(2,1)$–type coordinate) plus edge aggregation). Let $H=(V,E)$ with $V=\{x_1,x_2,x_3,x_4\}$ and

$$E=\left\{\{x_1,x_2,x_3\},\{x_2,x_3,x_4\}\right\}.$$

Its primal graph $G\left(H\right)$ has edges $\{x_1,x_2\}$, $\{x_1,x_3\}$, $\{x_2,x_3\}$, $\{x_2,x_4\}$, $\{x_3,x_4\}$. Choose $p=2$, $q=1$ with

$$L_V^{\left(1\right)}=\mathbb{Z}\left(\mathit{frequency}\mathit{channel}\right),L_V^{\left(2\right)}=\{0,1\}\left(\mathit{role}\right),L_E^{\left(1\right)}={\mathbb{Z}}_{\ge 0}\left(\mathit{edge}\mathit{cost}\right).$$

Define

$${\ell }_V^{\left(1\right)}\left(x_1\right)=0,{\ell }_V^{\left(1\right)}\left(x_2\right)=2,{\ell }_V^{\left(1\right)}\left(x_3\right)=5,{\ell }_V^{\left(1\right)}\left(x_4\right)=7,$$

$${\ell }_V^{\left(2\right)}\left(x_1\right)=0,{\ell }_V^{\left(2\right)}\left(x_2\right)=1,{\ell }_V^{\left(2\right)}\left(x_3\right)=0,{\ell }_V^{\left(2\right)}\left(x_4\right)=1,$$

$${\ell }_E^{\left(1\right)}\left(\{x_1,x_2,x_3\}\right)=5,{\ell }_E^{\left(1\right)}\left(\{x_2,x_3,x_4\}\right)=7.$$

Schema Φ requires:

On coordinate 1 (channels): for distinct $u,v\in V$,

$${\mathrm{dist}}_H(u,v)=1\Rightarrow |{\ell }_V^{\left(1\right)}\left(u\right)-{\ell }_V^{\left(1\right)}\left(v\right)|\ge 2,{\mathrm{dist}}_H(u,v)=2\Rightarrow |{\ell }_V^{\left(1\right)}\left(u\right)-{\ell }_V^{\left(1\right)}\left(v\right)|\ge 1.$$

(R) 

On coordinate 2 (roles): each hyperedge contains both roles 0 and 1.

(A) 

Edge aggregation: for each $e\in E$,

$${\ell }_E^{\left(1\right)}\left(e\right)=max\{{\ell }_V^{\left(1\right)}\left(v\right):v\in e\}.$$

Verification. Distances are taken in $G\left(H\right)$, which is the $K_4$ minus edges $\{x_1,x_4\}$; hence ${\mathrm{dist}}_H(x_1,x_4)=2$. All adjacent pairs satisfy channel gaps: $|0-2|=2$, $|0-5|=5$, $|2-5|=3$, $|2-7|=5$, $|5-7|=2$. Distance 2 pair $(x_1,x_4)$ has $|0-7|=7\ge 1$. (R) Each hyperedge $\{x_1,x_2,x_3\}$ and $\{x_2,x_3,x_4\}$ contains roles $\{0,1\}$. (A) For $\{x_1,x_2,x_3\}$ the max channel is $max\{0,2,5\}=5$, matching ${\ell }_E^{\left(1\right)}=5$; for $\{x_2,x_3,x_4\}$ the max is $max\{2,5,7\}=7$, matching ${\ell }_E^{\left(1\right)}=7$. Thus $({\ell }_V,{\ell }_E)$ satisfies Φ.

Theorem 4

(HyperGraph MultiLabeling generalizes hypergraph labeling and graph MultiLabeling). Let $(H,\mathsf{\Psi })$ be any (single–coordinate) hypergraph labeling problem, with $\mathsf{\Psi }(H,\lambda )$ a predicate over $\lambda :V\to L$ (or $\lambda :E\to L$) using only the hypergraph structure (incidence and/or ${\mathrm{dist}}_H$) and relations on L. Let $(G,{\mathsf{\Phi }}_{\mathrm{GML}})$ be any Graph MultiLabeling instance on a simple graph $G=(V,E_G)$ with p vertex and q edge coordinates.

Then:

• (Hypergraph labeling ⊆ HyperGraph MultiLabeling)  There exists a schema Φ with either $(p,q)=(1,0)$ (vertex case) or $(p,q)=(0,1)$ (edge case) such that

  $$\exists \lambda \mathrm{with}\mathsf{\Psi }(H,\lambda )⟺\exists ({\ell }_V,{\ell }_E)\mathrm{with}\mathsf{\Phi }\left(H;{\ell }_V,{\ell }_E\right),$$
  where ${\ell }_V^{\left(1\right)}=\lambda$ (or ${\ell }_E^{\left(1\right)}=\lambda$).
• (Graph MultiLabeling ⊆ HyperGraph MultiLabeling)  There exists a hypergraph $\widehat{H}=(V,\widehat{E})$ with $\widehat{E}:=\{\{u,v\}\subseteq V:uv\in E_G\}$ and a schema $\widehat{\mathsf{\Phi }}$ such that

  $$\exists ({\ell }_V,{\ell }_E)\mathrm{with}{\mathsf{\Phi }}_{\mathrm{GML}}\left(G;{\ell }_V,{\ell }_E\right)⟺\exists ({\ell }_V,{\ell }_E)\mathrm{with}\widehat{\mathsf{\Phi }}\left(\widehat{H};{\ell }_V,{\ell }_E\right).$$

Hence HyperGraph MultiLabeling strictly contains both classical hypergraph labeling and graph MultiLabeling.

Proof. (1) Vertex case. Take $p=1,q=0$, set $L_V^{\left(1\right)}:=L$, and define $\mathsf{\Phi }\left(H;\left({\ell }_V^{\left(1\right)}\right),⌀\right):=\mathsf{\Psi }\left(H,{\ell }_V^{\left(1\right)}\right)$. Then $\mathsf{\Psi }(H,\lambda )$ holds iff $\mathsf{\Phi }\left(H;\left(\lambda \right),⌀\right)$ holds; existence is equivalent. Edge case is identical with $p=0,q=1$ and $L_E^{\left(1\right)}:=L$.

(2) Let $\widehat{H}=(V,\widehat{E})$ be the 2–uniform hypergraph obtained from G by setting one hyperedge for each graph edge. Then $G\left(\widehat{H}\right)=G$, hence ${\mathrm{dist}}_{\widehat{H}}={\mathrm{dist}}_G$. Define $\widehat{\mathsf{\Phi }}(\widehat{H};{\ell }_V,{\ell }_E):={\mathsf{\Phi }}_{\mathrm{GML}}\left(G\left(\widehat{H}\right);{\ell }_V,{\ell }_E\right)$. Any constraint in ${\mathsf{\Phi }}_{\mathrm{GML}}$ that refers to adjacency/distances or incidence in G is identically interpreted in $\widehat{H}$ via $G\left(\widehat{H}\right)=G$, so the two existence statements are equivalent. □

2.5. SuperHyperGraph MultiLabeling

SuperHyperGraph MultiLabeling assigns multiple labels to supervertices and superedges, generalizing both hypergraph multilabeling and superhypergraph labeling for hierarchical multi-dimensional applications.

Definition 13 (Primal (2–section) of a SuperHyperGraph and distance). Let ${\mathrm{SHG}}^{\left(n\right)}=(V,E)$ be an n–SuperHyperGraph, i.e. $V\subseteq {\mathcal{P}}^n\left(V_0\right)$ and $⌀\ne E\subseteq \mathcal{P}\left(V\right)\setminus \{⌀\}$. Its primal graph (or 2–section) is

$$G\left({\mathrm{SHG}}^{\left(n\right)}\right):=(V,E^{\prime })\mathrm{with}{E}^{\prime }:=\left\{\{u,v\}\subseteq V|\exists F\in E:\{u,v\}\subseteq F\right\}.$$

The vertex distance ${\mathrm{dist}}_{\mathrm{SHG}}(x,y)$ is the usual shortest–path distance between $x,y\in V$ computed in $G\left({\mathrm{SHG}}^{\left(n\right)}\right)$.

Definition 14 (Base support (flattening)). For $n\ge 0$ define recursively a map ${flat}_n:{\mathcal{P}}^n\left(V_0\right)\to \mathcal{P}\left(V_0\right)$ by

$${flat}_0\left(v\right):=\left\{v\right\}(v\in V_0),{flat}_{n+1}\left(X\right):=\bigcup _{Y\in X}{flat}_n\left(Y\right)(X\in {\mathcal{P}}^{n+1}\left(V_0\right)).$$

Thus, for a supervertex $X\in V\subseteq {\mathcal{P}}^n\left(V_0\right)$, the set ${flat}_n\left(X\right)\subseteq V_0$ collects all base elements of $V_0$ that occur anywhere inside X.

Definition 15

(SuperHyperGraph MultiLabeling). Fix integers $p,q\ge 0$ and an n–SuperHyperGraph ${\mathrm{SHG}}^{\left(n\right)}=(V,E)$. Choose nonempty vertex label alphabets $L_V^{\left(1\right)},\dots ,L_V^{\left(p\right)}$ and nonempty edge label alphabets $L_E^{\left(1\right)},\dots ,L_E^{\left(q\right)}$. A SuperHyperGraph MultiLabeling consists of

$${\ell }_V=\left({\ell }_V^{\left(1\right)},\dots ,{\ell }_V^{\left(p\right)}\right),{\ell }_V^{\left(a\right)}:V\to L_V^{\left(a\right)}(1\le a\le p),$$

$${\ell }_E=\left({\ell }_E^{\left(1\right)},\dots ,{\ell }_E^{\left(q\right)}\right),{\ell }_E^{\left(b\right)}:E\to L_E^{\left(b\right)}(1\le b\le q).$$

Equivalently, a single vertex map ${\ell }_V:V\to {\prod }_{a=1}^p{L}_V^{\left(a\right)}$ and a single edge map ${\ell }_E:E\to {\prod }_{b=1}^q{L}_E^{\left(b\right)}$. A schema is a first–order predicate

$$\mathsf{\Phi }\left({\mathrm{SHG}}^{\left(n\right)};{\ell }_V,{\ell }_E\right)$$

built from the incidence relation $v\in F$ ($v\in V$, $F\in E$), the distance ${\mathrm{dist}}_{\mathrm{SHG}}$, the operators ${flat}_n$, basic set–theoretic/cardinality operations on $\mathcal{P}\left(V_0\right)$, and given relations/operations on the alphabets. The pair $({\ell }_V,{\ell }_E)$ is a valid SuperHyperGraph MultiLabeling (for $\mathsf{\Phi }$)if $\mathsf{\Phi }\left({\mathrm{SHG}}^{\left(n\right)};{\ell }_V,{\ell }_E\right)$ holds.

Remark 5

(Typical constraints that $\mathsf{\Phi }$ may express).

• Distance–aware separation on a vertex coordinate a:  if ${\mathrm{dist}}_{\mathrm{SHG}}(x,y)=1$ then $|{\ell }_V^{\left(a\right)}\left(x\right)-{\ell }_V^{\left(a\right)}\left(y\right)|\ge {\lambda }_1$, and if ${\mathrm{dist}}_{\mathrm{SHG}}(x,y)=2$ then $|{\ell }_V^{\left(a\right)}\left(x\right)-{\ell }_V^{\left(a\right)}\left(y\right)|\ge {\lambda }_2$.
• Support cardinality on a vertex coordinate a:  ${\ell }_V^{\left(a\right)}\left(X\right)=\left|{flat}_n\left(X\right)\right|$ for all $X\in V$.
• Superedge aggregation on an edge coordinate b:  ${\ell }_E^{\left(b\right)}\left(F\right)=g\left(\{{flat}_n\left(X\right):X\in F\}\right)$ for a fixed aggregator g (e.g. union size, intersection size, Jaccard index, maximum of a vertex coordinate, etc.).

Example 13

(A $(p,q)=(2,1)$ multilabel on a level-$n=1$ SuperHyperGraph). Let $V_0=\{a,b,c,d\}$. Consider $n=1$ so supervertices are subsets of $V_0$. Let

$$V=\left\{X_1,X_2,X_3\right\}:=\left\{\{a,b\},\{b,c\},\{c,d\}\right\},E=\left\{F_1,F_2\right\}:=\left\{\{X_1,X_2\},\{X_2,X_3\}\right\}.$$

Hence $G\left({\mathrm{SHG}}^{\left(1\right)}\right)$ is the path $X_1-X_2-X_3$. Choose alphabets

$$L_V^{\left(1\right)}=\{R,G,B\}\left(\mathit{colors}\right),L_V^{\left(2\right)}=\mathbb{N}\left(\mathit{support}\mathit{size}\right),L_E^{\left(1\right)}=\mathbb{N}\left(\mathit{coverage}\mathit{size}\right).$$

Define the labels

$${\ell }_V^{\left(1\right)}\left(X_1\right)=R,{\ell }_V^{\left(1\right)}\left(X_2\right)=G,{\ell }_V^{\left(1\right)}\left(X_3\right)=B,{\ell }_V^{\left(2\right)}\left(X_i\right)=\left|{flat}_1\left(X_i\right)\right|=\left|X_i\right|=2(i=1,2,3),$$

$${\ell }_E^{\left(1\right)}\left(F_1\right)=\left|{flat}_1\left(X_1\right)\cup {flat}_1\left(X_2\right)\right|=\left|\{a,b\}\cup \{b,c\}\right|=3,$$

$${\ell }_E^{\left(1\right)}\left(F_2\right)=\left|{flat}_1\left(X_2\right)\cup {flat}_1\left(X_3\right)\right|=\left|\{b,c\}\cup \{c,d\}\right|=3.$$

Schema Φ imposes simultaneously:

(Col)

Adjacent supervertices receive different colors on coordinate 1.

(Sup)

For all $X\in V$, ${\ell }_V^{\left(2\right)}\left(X\right)=\left|{flat}_1\left(X\right)\right|$.

(Agg)

For all $F\in E$, ${\ell }_E^{\left(1\right)}\left(F\right)=\left|{\bigcup }_{X\in F}{flat}_1\left(X\right)\right|$.

The given $\left({\ell }_V,{\ell }_E\right)$ satisfies (Col), (Sup), and (Agg), hence it is a valid SuperHyperGraph MultiLabeling.

Example 14

(A distance–aware multilabel on a level-$n=2$ SuperHyperGraph). Let $V_0=\{1,2,3\}$ and $n=2$. Define supervertices

$$A:=\left\{\left\{1\right\},\{1,2\}\right\},B:=\left\{\left\{2\right\},\{2,3\}\right\}\in {\mathcal{P}}^2\left(V_0\right),$$

and set $V=\{A,B\}$, $E=\left\{\{A,B\}\right\}$. Then

$${flat}_2\left(A\right)=\{1,2\},{flat}_2\left(B\right)=\{2,3\}.$$

Choose alphabets $L_V^{\left(1\right)}=\mathbb{Z}$ (channels), $L_V^{\left(2\right)}=\mathbb{N}$ (support size), $L_E^{\left(1\right)}=\mathbb{N}$ (overlap size). Define

$${\ell }_V^{\left(1\right)}\left(A\right)=0,{\ell }_V^{\left(1\right)}\left(B\right)=3\left(\mathit{gap}3\right),{\ell }_V^{\left(2\right)}\left(A\right)=2,{\ell }_V^{\left(2\right)}\left(B\right)=2,$$

$${\ell }_E^{\left(1\right)}\left(\{A,B\}\right)=\left|{flat}_2\left(A\right)\cap {flat}_2\left(B\right)\right|=\left|\{1,2\}\cap \{2,3\}\right|=1.$$

Let Φ require:

$L(2,1)$

On coordinate 1, if ${\mathrm{dist}}_{\mathrm{SHG}}(X,Y)=1$ then $|{\ell }_V^{\left(1\right)}\left(X\right)-{\ell }_V^{\left(1\right)}\left(Y\right)|\ge 2$ (here $|0-3|=3\ge 2$).

(Sup) 

${\ell }_V^{\left(2\right)}\left(X\right)=\left|{flat}_2\left(X\right)\right|$ for all $X\in V$ (true: 2 and 2).

(Int) 

${\ell }_E^{\left(1\right)}\left(F\right)=\left|{\bigcap }_{X\in F}{flat}_2\left(X\right)\right|$ for all $F\in E$ (true: 1).

Thus $({\ell }_V,{\ell }_E)$ satisfies Φ and is a valid multilabel.

Theorem 5

(SuperHyperGraph MultiLabeling generalizes SuperHyperGraph Labeling and HyperGraph MultiLabeling). The framework in Definition 15 strictly contains:

(i) 

SuperHyperGraph Labeling(single–coordinate labeling on supervertices and/or superedges);

(ii) 

HyperGraph MultiLabeling(multi–coordinate labeling on ordinary hypergraphs).

Proof. (i) Let a SuperHyperGraph labeling be given as a single map ${\lambda }_V:V\to L$ (vertex case) or ${\lambda }_E:E\to L$ (edge case), together with a predicate $\mathsf{\Psi }$ that uses only incidence, the primal distance and allowed relations on L. Take $p=1$, $q=0$ (vertex case) with $L_V^{\left(1\right)}:=L$, and set

$$\mathsf{\Phi }\left({\mathrm{SHG}}^{\left(n\right)};\left({\ell }_V^{\left(1\right)}\right),⌀\right):=\mathsf{\Psi }\left({\mathrm{SHG}}^{\left(n\right)};{\ell }_V^{\left(1\right)}\right).$$

Then ${\lambda }_V$ satisfies $\mathsf{\Psi }$ iff $({\ell }_V^{\left(1\right)},⌀)$ with ${\ell }_V^{\left(1\right)}={\lambda }_V$ satisfies $\mathsf{\Phi }$. The edge case is identical with $p=0$, $q=1$.

(ii) Let $H=(V,E)$ be a (finite) hypergraph and consider any HyperGraph MultiLabeling instance on H (with p vertex and q edge coordinates and a schema ${\mathsf{\Phi }}_{\mathrm{HG}}$ based on the hypergraph incidence/distance). Realize H as an $n=0$ SuperHyperGraph by taking $V_0:=V$ and ${\mathrm{SHG}}^{\left(0\right)}:=(V,E)$; then $G\left({\mathrm{SHG}}^{\left(0\right)}\right)$ coincides with the primal of H, so the same adjacency/distance is available. Define

$$\mathsf{\Phi }\left({\mathrm{SHG}}^{\left(0\right)};{\ell }_V,{\ell }_E\right):={\mathsf{\Phi }}_{\mathrm{HG}}\left(H;{\ell }_V,{\ell }_E\right).$$

Thus every feasible HyperGraph MultiLabeling on H is a feasible SuperHyperGraph MultiLabeling on ${\mathrm{SHG}}^{\left(0\right)}$, and conversely. Therefore the latter generalizes the former. □

3. Conclusion

In this paper, we defined and study the mathematical properties of Graph Labeling, HyperGraph Labeling, SuperHyperGraph Labeling, Graph MultiLabeling, HyperGraph MultiLabeling, and SuperHyperGraph MultiLabeling. We anticipate that future work may explore extensions employing Fuzzy Sets[33,34], Intuitionistic Fuzzy Sets[35,36], Neutrosophic Sets[17,37,38], Picture Fuzzy Sets [39,40], HyperFuzzy Sets [41,42,43], and Plithogenic Sets[44,45].