Chromatic Number and Neutrosophic Chromatic Number

New setting is introduced to study chromatic number. Neutrosophic chromatic number and 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 assigns to the vertices of neutrosophic graphs is applied. Some questions and problems are posed concerning ways to do further studies on this topic. Using strong edge 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 neutrosophic chromatic number. This specific relation amid edges is necessary to compute both chromatic number concerning the number of representative in the set of representatives and neutrosophic chromatic number concerning neutrosophic cardinality of set of representatives. If two vertices have no strong 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.

[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 It's motivation to find notions to use in any classes of neutrosophic graphs. 14 Real-world applications about time table and scheduling are another thoughts which 15 lead to be considered as motivation. Connections amid two items have key roles to 16 assign colors. Thus they're used to define new ideas which conclude to the structure of 17 coloring. The concept of having strong edge inspire to study the behavior of strong edge 18 in the way that, both neutrosophic chromatic number and chromatic number are the 19 cases of study. 20 The framework of this study is as follows. In section (1.2), I introduce basic 21 definitions to clarify about preliminaries. In section (2), new notion of coloring is 22 applied to the vertices of neutrosophic graphs. Neutrosophic strong edge has the key role 23 in this way. Classes of neutrosophic graphs are studied when the edges are neutrosophic 24 strong. In section (3), one application is posed for neutrosophic graphs concerning time 25 table and scheduling when the suspicions are about choosing some subjects. In section 26 (4), some problems and questions for further studies are proposed. In section (5), gentle 27 discussion about results and applications are featured. In section (5), a brief overview 28 concerning advantages and limitations of this study alongside conclusions are formed. is a subset of V × V such that this subset is symmetric. where it's complete and µ(xy) = σ(x) ∧ σ(y) for all xy ∈ E. 40 Definition 1.6. A neutrosophic graph G : (σ, µ) is called a neutrosophic strong 41 where µ(xy) = σ(x) ∧ σ(y) for all xy ∈ E. 42 Definition 1.7. A path v 0 , v 1 , · · · , v n is called neutrosophic path where 43 µ(v i v i+1 ) > 0, i = 0, 1, · · · , n − 1. i-path is a path with i edges, it's also called length 44 of path. 45 Definition 1.8. A crisp cycle v 0 , v 1 , · · · , v n , v 0 is called neutrosophic cycle where 46 there are two edges xy and uv such that µ(xy) = µ(uv) = i=0,1,··· ,n−1 µ(v i v i+1 ). 47 Definition 1.9. A neutrosophic graph is called neutrosophic t-partite if V is 48 partitioned to t parts, V 1 , V 2 , · · · , V t and the edge xy implies x ∈ V i and y ∈ V j where   Proof. All edges are neutrosophic strong. Every vertex has edge with n − 1 vertices.

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Thus n is chromatic number. Since any given vertex has different color in comparison to 82 another vertex, neutrosophic cardinality of V is neutrosophic chromatic number.

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Therefore, neutrosophic chromatic number is neutrosophic order. Proof. With alternative coloring on vertices, at end, two vertices have same color, and they've same edge. So, chromatic number is three. Since the colors are three, the vertices with minimum values in every color, are representatives. Hence, min x,y and z have different colors Proposition 2.7. Let N = (σ, µ) be a neutrosophic strong star with c as center. Then chromatic number is two and neutrosophic chromatic number is Proof. All edges are neutrosophic strong. Center vertex has common edge with every given vertex. So it has different color in comparison to other vertices. So one color has only one vertex which has that color. All non-center vertices have no common edge amid each other. Then they've same color. The representative of this color is a non-center vertex which has minimum value amid all non-center vertices. Hence, min x is non-center vertex Proposition 2.8. Let N = (σ, µ) be a neutrosophic strong wheel with c as center. Then chromatic number is three where neutrosophic cycle has even number as its length and neutrosophic chromatic number is Proof. Every given vertex has neutrosophic strong edge with all vertices from another part. So the color of every vertex which is in a same part is same. Hence, two parts implies two different colors. It induces chromatic number is two. The minimum value of a vertex amid all vertices in every part, identify the representative of every color. Therefore, min {σ(x 1 ) + σ(x 2 ) + · · · + σ(x t )}.
Proof. Every part has same color for its vertices. So chromatic number is t. Every part introduces one vertex as a representative of its color. Thus, neutrosophic chromatic number is min x1,x2 ,··· ,xt are in different parts {σ(x 1 ) + σ(x 2 ) + · · · + σ(x t )}.  Designing the programs to achieve some goals is general approach to apply on some 152 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 154 important to avoid mixing up.  Step 2. (Issue) scheduling of program has faced with difficulties to differ amid 159 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 161 data to assign every section and to assign to relation amid section, three numbers 162 belong unit interval to state indeterminacy, possibilities and determinacy. There's 163 one restriction in that, the numbers amid two section is at least the number of the 164 relation amid them.   This study uses mixed combinations of neutrosophic chromatic number and chromatic 198 number to study on neutrosophic graphs. The connections of vertices which are clarified 199 by neutrosophic strong edges, differ them from each other and and put them in different 200 categories to represent one representative for each color. Further studies could be about 201 changes in the settings to compare this notion amid different settings of graph theory. 202 One way is finding some relations amid array of vertices to make sensible defintions. In 203 Table (2), some limitations and advantages of this study is pointed out.