Dimension and Coloring alongside Domination in Neutrosophic Hypergraphs

New setting is introduced to study resolving number and chromatic number alongside dominating number. Different types of procedures including set, optimal set, and optimal number alongside study on the family of neutrosophic hypergraphs are proposed in this way, some results are obtained. General classes of neutrosophic hypergraphs are used to obtains these numbers and the representatives of the colors, dominating sets and resolving sets. Using colors to assign to the vertices of neutrosophic hypergraphs and characterizing resolving sets and dominating sets are applied. Some questions and problems are posed concerning ways to do further studies on this topic. Using different ways of study on neutrosophic hypergraphs to get new results about numbers and sets in the way that some numbers get understandable perspective. Family of neutrosophic hypergraphs are studied to investigate about the notions, dimension and coloring alongside domination in neutrosophic hypergraphs. In this way, sets of representatives of colors, resolving sets and dominating sets have key role. Optimal sets and optimal numbers have key points to get new results but in some cases, there are usages of sets and numbers instead of optimal ones. Simultaneously, three notions are applied into neutrosophic hypergraphs to get sensible results about their structures. Basic familiarities with neutrosophic hypergraphs theory and hypergraph theory are proposed for this article.


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(iii) : If S is set of all neutrosophic-dominating sets, then is called optimal-neutrosophic-dominating number and X is called 133 optimal-neutrosophic-dominating set. .

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(ii) : A set S is called neutrosophic-resolving set if for every y ∈ V \ S, there's 137 at least one vertex x which neutrosophic-resolves vertices y, w.

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(iii) : If S is set of all neutrosophic-resolving sets, then is called optimal-neutrosophic-resolving number and X is called 139 optimal-neutrosophic-resolving set.
is called optimal-neutrosophic-coloring number and X is called 146 optimal-neutrosophic-coloring set.      Proof. Suppose neutrosophic hypergraph N HG = (V, E, σ, µ). Consider optimal-neutrosophic-coloring number is It implies there's one neutrosophic hyperedge which has all neutrosophic vertices. Since 207 if all neutrosophic vertices are incident to a neutrosophic hyperedge, then all have 208 different colors.

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Proposition 3.4. Assume neutrosophic hypergraph N HG = (V, E, σ, µ). If there's at least one hyperedge which contains n vertices where n is the cardinality of the set V, then optimal-neutrosophic-coloring number is Proof. Consider neutrosophic hypergraph N HG = (V, E, σ, µ). Suppose there's at least one hyperedge which contains n vertices where n is the cardinality of the set V. It implies there's one neutrosophic hyperedge which has all neutrosophic vertices. If all neutrosophic vertices are incident to a neutrosophic hyperedge, then all have different colors. So V is optimal-neutrosophic-coloring set. It induces optimal-neutrosophic-coloring number is Proposition 3.5. Assume neutrosophic hypergraph N HG = (V, E, σ, µ). If optimal-neutrosophic-dominating number is then there's at least one neutrosophic vertex which doesn't have incident to any 211 neutrosophic hyperedge.

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Proof. Suppose neutrosophic hypergraph N HG = (V, E, σ, µ). Consider optimal-neutrosophic-dominating number is If for all given neutrosophic vertex, there's at least one neutrosophic hyperedge which the neutrosophic vertex has incident to it, then there's a neutrosophic vertex x such that optimal-neutrosophic-dominating number is It induces contradiction with hypothesis. It implies there's at least one neutrosophic 213 vertex which doesn't have incident to any neutrosophic hyperedge.
It implies every neutrosophic vertex isn't neutrosophic-resolved by a neutrosophic 217 vertex. It's contradiction with hypothesis. So every given vertex doesn't have incident 218 to any hyperedge.

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Proposition 3.9. Assume neutrosophic hypergraph N HG = (V, E, σ, µ). If optimal-neutrosophic-coloring number is then all neutrosophic verties which have incident to at least one neutrosophic hyperedge. 221 Proof. Suppose neutrosophic hypergraph N HG = (V, E, σ, µ). Consider optimal-neutrosophic-coloring number is If for all given neutrosophic vertices, there's no neutrosophic hyperedge which the neutrosophic vertices have incident to it, then there's neutrosophic vertex x such that optimal-neutrosophic-coloring number is It induces contradiction with hypothesis. It implies all neutrosophic vertices have 222 incident to at least one neutrosophic hyperedge.
Proof. Consider neutrosophic hypergraph N HG = (V, E, σ, µ). Thus V − {x} isn't a neutrosophic-coloring set. Since if not, x isn't incident to any given neutrosophic hyperedge. This is contradiction with supposition. It induces that x belongs to a neutrosophic hyperedge which has another vertex s. It implies s neutrosophic-colors x. Thus V − {x} isn't a neutrosophic-coloring set. It induces optimal-neutrosophic-coloring number isn't < Σ v∈V σ(v).

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Proposition 3.11. Assume neutrosophic hypergraph N HG = (V, E, σ, µ). Then 225 optimal-neutrosophic-dominating set has cardinality which is greater than n − 1 where n 226 is is the cardinality of the set V.
Proof. Consider neutrosophic hypergraph N HG = (V, E, σ, µ). The set V is 228 neutrosophic-dominating set. So optimal-neutrosophic-dominating set has cardinality 229 which is greater than n where n is is the cardinality of the set V. But the set V \ {x}, 230 for every given neutrosophic vertex is optimal-neutrosophic-dominating set has 231 cardinality which is greater than n − 1 where n is is the cardinality of the set V. The 232 result is obtained.

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Proposition 3.12. Assume neutrosophic hypergraph N HG = (V, E, σ, µ). S is maximum set of vertices which form a hyperedge. Then S is optimal-neutrosophic-coloring set and is optimal-neutrosophic-coloring number. In other hand, S is maximum set of vertices which form a hyperedge. It induces optimal-neutrosophic-coloring number ≤ Σ s∈S σ(S).
So S is neutrosophic-coloring set. Hence S is optimal-neutrosophic-coloring set and is optimal-neutrosophic-coloring number.  Designing the programs to achieve some goals is general approach to apply on some 409 issues to function properly. Separation has key role in the context of this style.

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Separating the duration of work which are consecutive, is the matter and it has 411 important to avoid mixing up.  Step 2. (Issue) Scheduling of program has faced with difficulties to differ amid 416 consecutive section. Beyond that, sometimes sections are not the same.

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Step 3. (Model) As Figure (5), the situation is designed as a model. The model uses 418 data to assign every section and to assign to relation amid section, three numbers 419 belong unit interval to state indeterminacy, possibilities and determinacy. There's 420 one restriction in that, the numbers amid two sections are at least the number of 421 the relation amid them. Table (1), clarifies about the assigned numbers to these 422 situation.  Figure 5. Vertices are suspicions about choosing them.
Step 4. (Solution) As Figure (5) shows, neutrosophic hyper graph as model, 424 proposes to use different types of coloring, resolving and dominating as numbers, 425 sets, optimal numbers, optimal sets and et cetera.