Neutrosophic Chromatic Number Based on Connectedness

New setting is introduced to study chromatic number. vital chromatic number and n-vital chromatic number are proposed in this way, some results are obtained. Classes of neutrosophic graphs are used to obtains these numbers and the representatives of the colors. Using colors to assign to the vertices of neutrosophic graphs is applied. Some questions and problems are posed concerning ways to do further studies on this topic. Using vital edge from connectedness to define the relation amid vertices which implies having different colors amid them and as consequences, choosing one vertex as a representative of each color to use them in a set of representatives and finally, using neutrosophic cardinality of this set to compute vital chromatic number. This specific relation amid edges is necessary to compute both vital chromatic number concerning the number of representative in the set of representatives and n-vital chromatic number concerning neutrosophic cardinality of set of representatives. If two vertices have no vital edge, then they can be assigned to same color even they’ve common edge. Basic familiarities with neutrosophic graph theory and graph theory are proposed for this article.

[3], studies on neutrosophic graphs in Ref. [1], relevant definitions of other 4 graphs based on fuzzy graphs in Ref. [12], related definitions of other graphs based on 5 neutrosophic graphs in Ref.
[4], are proposed. 6 In this section, I use two subsections to illustrate a perspective about the 7 background of this study.  In this study, there's an idea which could be considered as a motivation. 10 Question 1.1. Is it possible to use mixed versions of ideas concerning "connectedness", 11 "neutrosophic graphs" and "neutrosophic coloring" to define some notions which are Definition 1.11. Let G : (σ, µ) be a neutrosophic graph. Neutrosophic cardinality of 60 V is called neutrosophic order of G and it's denoted by O n (G).    Figure 1. Neutrosophic graph N 1 is considered with respect to first order. It's complete but it isn't neutrosophic complete. It's cycle but it isn't neutrosophic cycle. It's neutrosophic 3-partite but it isn't neutrosophic complete 3-partite.
Proof. Consider N = (σ, µ) be a neutrosophic cycle. Hence, there are at least two edges which are weakest, it means there are xy, uv ∈ E such that In other hand, for every given vertices x and y, there are two paths from x to y. So for every given path, S(P ) = min e∈E µ(e).
Thus for every x, y ∈ V, xy ∈ E, the value µ(xy) forms the connectedness amid x to y.

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Therefore connectedness amid any given couple of vertices, doesn't change when they 94 form an edge and they're deleted. It induces every edge is vital.

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Proof. Suppose N = (σ, µ) is a neutrosophic complete which is neither neutrosophic empty nor neutrosophic path. If x, y ∈ V, then xy ∈ E. Thus P : x, y is a path for every given couple of vertices. Hence S(P ) = µ(xy).

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Proposition 3.3. Let N = (σ, µ) be a neutrosophic graph which is fixed-edge and 100 which is neither neutrosophic empty nor neutrosophic path. Then all edges are vital.
It induces for every given edge e and every given paths P, P S(P ) = S(P ) = µ(e).
It implies connectedness is fixed and it equals to µ(e) where e ∈ E. Therefore, the  Proof. Consider N = (σ, µ) is a neutrosophic graph which is neither neutrosophic empty nor neutrosophic path. Assume N = (σ, µ) is a neutrosophic graph which is either fixed-edge or fixed-vertex and neutrosophic strong. Hence, all edges have same value. It means ∀e, e ∈ E, µ(e) = µ(e ).
It induces for every given edge e and every given paths P, P S(P ) = S(P ) = µ(e).
It implies connectedness is fixed and it equals to µ(e) where e ∈ E. Therefore, the deletion of e has no change on connectedness amid every couple of vertices. It means every edge is vital. In other hand, suppose otherwise. So by |E| > 2, there's one edge e such that for every edge e = e, µ(e) > µ(e ).
Let a number µ(e ) be min e∈E µ(e).
Then connectedness is ≥ µ(e ). But there's a cycle which implies |E| > 3. It induces 107 there are at least two paths corresponded to e . By µ(e) > µ(e ), connectedness ≥ µ(e ). 108 It implies corresponded connectedness to e isn't changed when the deletion of e is 109 done. Thus the edge e ∈ E is vital.

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Proposition 3.5. Let N = (σ, µ) be a neutrosophic strong which is fixed-vertex and 111 which is neither neutrosophic empty nor neutrosophic path. Then all edges are vital.
It induces for every couple of vertices which form an edge, connectedness amid them is 113 same and equals µ(e) where e is a given edge. It implies at least there are two paths 114 with strength µ(e). Thus deletion of every edge has no change on connectedness amid 115 its vertices. Therefore, every edge is vital.   And at most min x,y,z∈V,xy,yz,xz∈E σ(x) + σ(y) + σ(z).
Proof. Suppose N = (σ, µ) is a neutrosophic cycle. By using alternative coloring of vertices, two or three numbers of colors are used. So the cardinality of set of representative is two or three. There are only these possibilities. Therefore n-vital chromatic number is at least min x,y∈V, xy∈E σ(x) + σ(y).

And at most min
x,y,z∈V,xy,yz,xz∈E σ(x) + σ(y) + σ(z). Proof. Assume N = (σ, µ) is neutrosophic bipartite which is fixed-edge and complete. It's fixed-edge so all edges have same value and as its consequences, all paths have same strength and all connectedness are same. Hence all edges are vital. By it's complete, all vertices from one part are connected to all vertices from another part. By it's bipartite, there are two colors to use on vertices such that every part has same color. So the set of representatives has the cardinality two which implies n-vital chromatic number is min x,y∈V, xy∈E σ(x) + σ(y).
Proof. Assume N = (σ, µ) is neutrosophic t−partite which is fixed-edge and complete. All parts have same color on their vertices. By it's fixed-edge and applying Proposition (3.7), all edges are vital. Thus minimum number of colors is t. And the set of representatives has the cardinality t. It means n-vital chromatic number is min x1,x2,··· ,xt∈V, xixj ∈E σ(x 1 ) + σ(x 2 ) + · · · + σ(x t ).
Proof. Consider N = (σ, µ) is neutrosophic wheel which is fixed-vertex and neutrosophic strong. By fixed-vertex and neutrosophic strong, it's fixed-edge. By it's fixed-edge and applying Proposition (3.7), all edges are vital. Center is connected to non-center vertices. So center uses unique color. Non-center vertices form a cycle. If the cycle is even, then n-vital chromatic number is min y,z∈V,yz∈E σ(c) + σ(y) + σ(z).

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The relation amid neutrosophic chromatic number and main parameters of  Time Table and Scheduling   278 Designing the programs to achieve some goals is general approach to apply on some 279 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 281 important to avoid mixing up.
Step 1. (Definition) Time table is an approach to get some attributes to do the 283 work fast and proper. The style of scheduling implies special attention to the 284 tasks which are consecutive.

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Step 2. (Issue) Scheduling of program has faced with difficulties to differ amid 286 consecutive section. Beyond that, sometimes sections are not the same.

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Step 3. (Model) As Figure (2), the situation is designed as a model. The model uses 288 data to assign every section and to assign to relation amid section, three numbers 289 belong unit interval to state indeterminacy, possibilities and determinacy. There's 290 one restriction in that, the numbers amid two sections are at least the number of 291 the relation amid them. Table (1), clarifies about the assigned numbers to these 292 situation.    Table (2), some limitations and advantages of this study is pointed out.