3.32 Optimal-super-coloring Number in n-SuperHyperGraph

3. Neutrosophic Hypergraphs

Proof. Consider n-SuperHyperGraph n-S H G = (Gn ⊆ P n(V ), En ⊆ P n(V )). Thus Gn − {Xn}, is a super-dominating set where Xn ∈ Gn . Since if not, Xn isn’t incident to any given super-edge. This is contradiction with supposition. It induces that Xn belongs to a super-edge which has another super-vertex X n. It implies X n super-dominates Xn. Thus Gn − {Xn} is a super-dominating set. It induces optimal-super-dominating number is <

|V |.

Proposition 3.26.6. Assume n-SuperHyperGraph n-S H G = (Gn ⊆ P n(V ), En ⊆ P n(V )). If optimal-super-resolving number is

|V |.

then every given super-vertex doesn’t have incident to any super-edge.

Proof. Consider n-SuperHyperGraph n-S H G = (Gn ⊆ P n(V ), En ⊆ P n(V )). Let optimal-super-resolving number be

|V |.

If it implies there’s a super-vertex is super-resolved by a super-vertex, then it’s contradiction with hypothesis. So every given super-vertex doesn’t have incident to any super-edge. Proposition 3.26.7. Assume n-SuperHyperGraph n-S H G = (Gn ⊆ P n(V ), En ⊆ P n(V )). Then optimal-super-resolving number is <

|V |.

Proof. Consider n-SuperHyperGraph n-S H G = (Gn ⊆ P n(V ), En ⊆ P n(V )). If optimal-super-resolving number is

|V |,

then there’s a contradiction to hypothesis. Since the set Gn − {Xn}, is superresolving set. It implies optimal-super-resolving number is <

|V |.

Proposition 3.26.8. Assume n-SuperHyperGraph n-S H G = (Gn ⊆ P n(V ), En ⊆ P n(V )). If optimal-super-coloring number is

|V |,

then al l super-vertices which have incident to at least one super-edge.

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3.26. Optimal Numbers For n-SuperHyperGraph

Proof. Suppose n-SuperHyperGraph n-S H G = (Gn ⊆ P n(V ), En ⊆ P n(V )). Consider optimal-super-coloring number is

|V | − |Xn|.

If for all given super-vertices, there’s no super-edge which the super-vertices have incident to it, then there’s super-vertex Xn such that optimal-super-coloring number is

|V | − |Xn|. It induces contradiction with hypothesis. It implies all super-vertices have incident to at least one super-edge. Proposition 3.26.9. Assume n-SuperHyperGraph n-S H G = (Gn ⊆ P n(V ), En ⊆ P n(V )). Then optimal-super-coloring number isn’t <

|V |.

Proof. Consider n-SuperHyperGraph n-S H G = (Gn ⊆ P n(V ), En ⊆ P n(V )). Thus Gn − {Xn} isn’t a super-coloring set. Since if not, Xn isn’t incident to any given super-edge. This is contradiction with supposition. It induces that Xn belongs to a super-edge which has another super-vertex Sn. It implies Sn super-colors Xn. Thus Gn − {Xn} isn’t a super-coloring set. It induces optimal-super-coloring number isn’t <

|V |.

Proposition 3.26.10. Assume n-SuperHyperGraph n-S H G = (Gn ⊆ P n(V ), En ⊆ P n(V )). Then optimal-super-dominating set has cardinality which is greater than n − 1 where n is the cardinality of the set V .

Proof. Consider n-SuperHyperGraph n-S H G = (Gn ⊆ P n(V ), En ⊆ P n(V )). The set Gn is super-dominating set. So optimal-super-dominating set has cardinality which is greater than n − 1 where n is the cardinality of the set V . But the set Gn \ {Xn}, for every given super-vertex Xn is optimal-superdominating set has cardinality which is greater than n − 1 where n is is the cardinality of the set V . The result is obtained. Proposition 3.26.11. Assume n-SuperHyperGraph n-S H G = (Gn ⊆ P n(V ), En ⊆ P n(V )). S is maximum set of super-vertices which form a super-edge. Then S is optimal-super-coloring set and

|{∪Xn | Xn ∈ S}| is optimal-super-coloring number.

Proof. Suppose n-SuperHyperGraph n-S H G = (Gn ⊆ P n(V ), En ⊆ P n(V )). Consider S is maximum set of super-vertices which form a super-edge. Thus all super-vertices of S have incident to super-edge. It implies the number of different colors equals to a cardinality based on S. Therefore, optimal-super-coloring number ≥

|{∪Xn | Xn ∈ S}|

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3. Neutrosophic Hypergraphs

In other hand, S is maximum set of super-vertices which form a super-edge. It induces optimal-super-coloring number ≤ |{∪Xn | Xn ∈ S}| So S is super-coloring set. Hence S is optimal-super-coloring set and

|{∪Xn | Xn ∈ S}| is optimal-super-coloring number.

3.27 Optimal Sets For n-SuperHyperGraph

Proposition 3.27.1. Assume n-SuperHyperGraph n-S H G = (Gn ⊆ P n(V ), En ⊆ P n(V )). If S is super-dominating set, then D contains S is super-dominating set.

Proof. Consider n-SuperHyperGraph n-S H G = (Gn ⊆ P n(V ), En ⊆ P n(V )). Suppose S is super-dominating set. Then all super-vertices are super-dominated. Thus D contains S is super-dominating set. Proposition 3.27.2. Assume n-SuperHyperGraph n-S H G = (Gn ⊆ P n(V ), En ⊆ P n(V )). If S is super-resolving set, then D contains S is super-resolving set.

Proof. Suppose n-SuperHyperGraph n-S H G = (Gn ⊆ P n(V ), En ⊆ P n(V )). Consider S is super-resolving set. Hence All two given super-vertices are super-resolved by at least one super-vertex of S. It induces D contains S is super-resolving set. Proposition 3.27.3. Assume n-SuperHyperGraph n-S H G = (Gn ⊆ P n(V ), En ⊆ P n(V )). If S is super-coloring set, then D contains S is supercoloring set.

Proof. Suppose n-SuperHyperGraph n-S H G = (Gn ⊆ P n(V ), En ⊆ P n(V )). Consider S is super-coloring set. So all super-vertices which have a common super-edge have different colors. Thus every super-vertex super-colored by a super-vertex of S. It induces every super-vertex which has a common super-edge has different colors with other super-vertices belong to that super-edge. Then D contains S is super-coloring set. Proposition 3.27.4. Assume n-SuperHyperGraph n-S H G = (Gn ⊆ P n(V ), En ⊆ P n(V )). Then Gn is super-dominating set.

Proof. Suppose n-SuperHyperGraph n-S H G = (Gn ⊆ P n(V ), En ⊆ P n(V )). Since Gn \ {Xn} is super-dominating set. Then Gn contains Gn \ {Xn} is super-dominating set. Proposition 3.27.5. Assume n-SuperHyperGraph n-S H G = (Gn ⊆ P n(V ), En ⊆ P n(V )).Then Gn is super-resolving set.

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