Different Types of Neutrosophic Chromatic Number

New setting is introduced to study chromatic number. Different types of chromatic numbers and neutrosophic 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 different types of edges from connectedness in same neutrosophic graphs and in modified neutrosophic graphs 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 types of chromatic numbers. This specific relation amid edges is necessary to compute both types of chromatic number concerning the number of representative in the set of representatives and types of neutrosophic chromatic number concerning neutrosophic cardinality of set of representatives. If two vertices have no intended 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.

definitions to clarify about preliminaries. In section "New Ideas", new notion of coloring 22 is applied to the vertices of neutrosophic graphs. Specific edge from connectedness has 23 the key role in this way. Classes of neutrosophic graphs are studied in the terms of 24 different types of edges in section "New Results". In section "Applications in Time 25 Table and Scheduling", one application is posed for neutrosophic graphs concerning  Proposition 2.6. Let N = (σ, µ) be a neutrosophic graph which is strong fixed-vertex. 97 Then N = (σ, µ) is fixed-edge. 98 Proposition 2.7. Let N = (σ, µ) be a neutrosophic graph which is strong fixed-vertex. 99 Then all edges are type-I. 100 Example 2.8. Consider Figure (2). All edges are type-I.

107
(i) : From n 1 to n 2 , there's no edge which is type-II but n 1 n 2 . 108 (ii) : From n 2 to n 3 , there's no edge which is type-II but n 1 n 2 . 109 (iii) : From n 1 to n 3 , there's no edge which is type-II but n 1 n 2 and n 2 n 3 . 110 (b) : Consider Figure (2). There's no edge which is type-II. 111 Definition 2.11. Let N = (σ, µ) be a neutrosophic graph. A neutrosophic edge xy is 112 called type-III if value of xy is the only value which is connectedness which is a 113 maximum strength of paths amid them. 114 Example 2.12. The comparison amid the variant of edges which are either type-I or 115 type-II or type-III, is possible when common neutrosophic graphs are studied.

117
(i) : From n 1 to n 2 , there's no edge which is type-III but n 2 n 3 . 118 (ii) : From n 2 to n 3 , there's no edge which is type-III but n 2 n 3 . 119 (iii) : From n 1 to n 3 , there's no edge which is type-III but n 1 n 3 and n 2 n 3 . Definition 2.13. Let N = (σ, µ) be a neutrosophic graph. A neutrosophic edge xy is 122 called type-IV if value of xy is connectedness which is a maximum strength of paths 123 amid them but in N = (σ, µ) doesn't have xy. 124 Example 2.14. The comparison amid the variant of edges which are either type-I 125 or...or type-IV, is possible when common neutrosophic graphs are studied.

147
(i) : From n 1 to n 2 , there's no edge which is type-VI. 148 (ii) : From n 2 to n 3 , edges n 2 n 3 and n 1 n 3 are type-VI. 149 (iii) : From n 1 to n 3 , edges n 2 n 3 and n 1 n 3 are type-VI.  Common way to define the number, could be twofold. One is about the cardinality 162 and another is about neutrosophic cardinality.

163
Definition 2.21. Let N = (σ, µ) be a neutrosophic graph. A vertex which has 164 common type edge with another vertex, has assigned different color from that vertex.

165
The cardinality of the set of representatives of colors, is called type chromatic 166 number and its neutrosophic cardinality concerning the set of representatives of colors 167 is called n-type chromatic number.

168
Definition 2.22. It's worthy to note that there are two types of definitions. One is 169 about the comparison amid edges and connectedness. Another is about one edge when 170 it's deleted, new connectedness is compared to deleted edge. Thus in first type, all edges 171 are compared to connectedness but in second type, for every edge, there's a 172 computation to have connectedness. So in first type, connectedness is unique and 173 there's one number for all edges as connectedness but in second type, for every edge, 174 there's a new connectedness to decide about the edge whether has intended attribute or 175 not. To avoid confusion, chromatic number is computed with respect to n 1 and n 2 176 where second style is used and all edges are labelled even they're not deleted edges so 177 third type is introduced when deletion of one edge, is enough to label all edges. Also 178 first order is used to have these concepts.

179
In following example, third type of definitions which are except from 180 type-IV,V,VI,VII, are studied.

181
Example 2.23. The comparison amid the variant of numbers which are either type-I 182 or...or type-VII, is possible when common neutrosophic graphs are studied. Chromatic 183 number is computed with respect to n 1 and n 2 . Also first order is used to have these number is 2 and n-type-I chromatic number is 1.73. (ii) : The set of representatives of colors is {n 1 , n 2 }. Thus type-II chromatic 189 number is 2 and n-type-II chromatic number is 1.73. 190 (iii) : The set of representatives of colors is {n 2 , n 3 }. Thus type-III chromatic 191 number is 2 and n-type-III chromatic number is 1.28. 192 (iv) : The set of representatives of colors is {n 2 , n 3 }. Thus type-IV chromatic 193 number is 2 and n-type-IV chromatic number is 1.28. 194 (v) : The set of representatives of colors is {n 1 , n 2 }. Thus type-V chromatic 195 number is 2 and n-type-V chromatic number is 1.73. 196 (vi) : The set of representatives of colors is {n 2 , n 3 }. Thus type-VI chromatic 197 number is 2 and n-type-VI chromatic number is 1.28. 198 (vii) : The set of representatives of colors is {n 2 , n 3 }. Thus type-VII chromatic 199 number is 2 and n-type-VII chromatic number is 1.28. 239 (ii). All edges have same amount so the connectedness amid two given edges is the 240 same. All edges aren't type-II. By it's neutrosophic complete, every vertex has n − 1 241 vertices which have common edges which aren't type-II. Thus the set of representatives 242 of colors is {}. The type-II chromatic number is 0 and n-type-II chromatic number is 0. 243 (iii). All edges have same amount so the connectedness amid two given edges is the 244 same. All edges aren't type-III. By it's neutrosophic complete, every vertex has n − 1 245 vertices which have common edges which aren't type-III. Thus the set of representatives 246 of colors is {}. The type-III chromatic number is 0 and n-type-III chromatic number is 0. 247 (iv). All edges have same amount so the connectedness amid two given edges is the 248 same. All edges are type-IV. By it's neutrosophic complete, every vertex has n − 1 249 vertices which have common edges which are type-IV. Thus the set of representatives of 250 colors is {v 1 , v 2 , · · · , v n }. The type-IV chromatic number is n and n-type-IV chromatic 251 number is neutrosophic cardinality of V. 252 (v). All edges have same amount so the connectedness amid two given edges is the 253 same. All edges aren't type-V. By it's neutrosophic complete, every vertex has n − 1 254 vertices which have common edges which aren't type-V. Thus the set of representatives 255 of colors is {}. The type-V chromatic number is 0 and n-type-V chromatic number is 0. 256 (vi). All edges have same amount so the connectedness amid two given edges is the 257 same. All edges aren't type-VI. By it's neutrosophic complete, every vertex has n − 1 258 vertices which have common edges which aren't type-VI. Thus the set of representatives 259 of colors is {}. The type-VI chromatic number is 0 and n-type-VI chromatic number is 0. 260 (vii). All edges have same amount so the connectedness amid two given edges is the 261 same. All edges aren't type-VII. By it's neutrosophic complete, every vertex has n − 1 262 vertices which have common edges which aren't type-VII. Thus the set of 263 representatives of colors is {}. The type-VII chromatic number is 0 and n-type-VII 264 chromatic number is 0.  which are type-I. Thus the set of representatives of colors is {v 1 , v 2 , · · · , v n }. The type-I 286 chromatic number is n and n-type-I chromatic number is nσ(v i ).

188
287 (ii). By it's fixed-vertex and it's neutrosophic complete, all edges have same amount 288 so the connectedness amid two given edges is the same. All edges aren't type-II. By it's 289 neutrosophic complete, every vertex has n − 1 vertices which have common edges which 290 aren't type-II. Thus the set of representatives of colors is {}. The type-II chromatic 291 number is 0 and n-type-II chromatic number is 0. 292 (iii). By it's fixed-vertex and it's neutrosophic complete, all edges have same amount 293 so the connectedness amid two given edges is the same. All edges aren't type-III. By it's 294 neutrosophic complete, every vertex has n − 1 vertices which have common edges which 295 aren't type-III. Thus the set of representatives of colors is {}. The type-III chromatic 296 number is 0 and n-type-III chromatic number is 0. 297 (iv). By it's fixed-vertex and it's neutrosophic complete, all edges have same amount 298 so the connectedness amid two given edges is the same. All edges are type-IV. By it's 299 neutrosophic complete, every vertex has n − 1 vertices which have common edges which 300 are type-IV. Thus the set of representatives of colors is {v 1 , v 2 , · · · , v n }. The type-IV 301 chromatic number is n and n-type-IV chromatic number is nσ (v i (v). By it's fixed-vertex and it's neutrosophic complete, all edges have same amount 303 so the connectedness amid two given edges is the same. All edges aren't type-V. By it's 304 neutrosophic complete, every vertex has n − 1 vertices which have common edges which 305 aren't type-V. Thus the set of representatives of colors is {}. The type-V chromatic 306 number is 0 and n-type-V chromatic number is 0. 307 (vi). By it's fixed-vertex and it's neutrosophic complete, all edges have same amount 308 so the connectedness amid two given edges is the same. All edges aren't type-VI. By it's 309 neutrosophic complete, every vertex has n − 1 vertices which have common edges which 310 aren't type-VI. Thus the set of representatives of colors is {}. The type-VI chromatic 311 number is 0 and n-type-VI chromatic number is 0. 312 (vii). By it's fixed-vertex and it's neutrosophic complete, all edges have same 313 amount so the connectedness amid two given edges is the same. All edges aren't 314 type-VII. By it's neutrosophic complete, every vertex has n − 1 vertices which have 315 common edges which aren't type-VII. Thus the set of representatives of colors is {}.

316
The type-VII chromatic number is 0 and n-type-VII chromatic number is 0.
The type-I chromatic number is t and 340 n-type-I chromatic number is neutrosophic cardinality of {v 1 , v 2 , · · · , v t }. 341 (ii). All edges have same amount so the connectedness amid two given edges is the 342 same. All edges aren't type-II. By it's neutrosophic strong, there's a vertex has   (iv). All edges have same amount so the connectedness amid two given edges is the 352 same. All edges are type-IV. By it's neutrosophic strong, there's a vertex has t = ∆(N ) 353 vertices which have common edges which are type-IV. Thus the set of representatives of 354 colors is {v 1 , v 2 , · · · , v t }. The type-IV chromatic number is t and n-type-IV chromatic 355 number is neutrosophic cardinality of {v 1 , v 2 , · · · , v t }. 356 (v). All edges have same amount so the connectedness amid two given edges is the   (vii). All edges have same amount so the connectedness amid two given edges is the 367 same. All edges aren't type-VII. By it's neutrosophic strong, there's a vertex has type-I chromatic number is t and n-type-I chromatic number is tσ (v i type-IV chromatic number is t and n-type-IV chromatic number is tσ(v i ).
The type-I 391 chromatic number is t and n-type-I chromatic number is tσ(v i ).
392 (ii). By it's fixed-vertex and it's neutrosophic strong, all edges have same amount so 393 the connectedness amid two given edges is the same. All edges aren't type-II. By it's 394 neutrosophic strong, there's a vertex has t = ∆(N ) vertices which have common edges 395 which aren't type-II. Thus the set of representatives of colors is {}. The type-II 396 chromatic number is 0 and n-type-II chromatic number is 0. 397 (iii). By it's fixed-vertex and it's neutrosophic strong, all edges have same amount 398 so the connectedness amid two given edges is the same. All edges aren't type-III. By it's 399 neutrosophic strong, there's a vertex has t = ∆(N ) vertices which have common edges 400 which aren't type-III. Thus the set of representatives of colors is {}. The type-III 401 chromatic number is 0 and n-type-III chromatic number is 0. 402 (iv). By it's fixed-vertex and it's neutrosophic strong, all edges have same amount so 403 the connectedness amid two given edges is the same. All edges are type-IV. By it's 404 neutrosophic strong, there's a vertex has t = ∆(N ) vertices which have common edges 405 which are type-IV. Thus the set of representatives of colors is {v 1 , v 2 , · · · , v t }. The 406 type-IV chromatic number is t and n-type-IV chromatic number is tσ (v i (v). By it's fixed-vertex and it's neutrosophic strong, all edges have same amount so 408 the connectedness amid two given edges is the same. All edges aren't type-V. By it's   (vii). By it's fixed-vertex and it's neutrosophic strong, all edges have same amount 418 so the connectedness amid two given edges is the same. All edges aren't type-VII. By  ii). All edges have same amount so the connectedness amid two given edges is the 444 same. All edges aren't type-II. By it's neutrosophic strong, there's a vertex has 2 445 vertices which have common edges which aren't type-II. Thus the set of representatives 446 of colors is {}. The type-II chromatic number is 0 and n-type-II chromatic number is 0. 447 (iii). All edges have same amount so the connectedness amid two given edges is the 448 same. All edges aren't type-III. By it's neutrosophic strong, there's a vertex has 2 449 vertices which have common edges which aren't type-III. Thus the set of representatives 450 of colors is {}. The type-III chromatic number is 0 and n-type-III chromatic number is 0. 451 (iv). All edges have same amount so the connectedness amid two given edges is the 452 same. All edges aren't type-IV. Since it's impossible to define when there's no cycle in 453 neutrosophic graph. 454 (v). All edges have same amount so the connectedness amid two given edges is the 455 same. All edges aren't type-V. Since it's impossible to define when there's no cycle in 456 neutrosophic graph. 457 (vi). All edges have same amount so the connectedness amid two given edges is the 458 same. All edges aren't type-VI. Since it's impossible to define when there's no cycle in 459 neutrosophic graph. 460 (vii). All edges have same amount so the connectedness amid two given edges is the 461 same. All edges aren't type-VII. Since it's impossible to define when there's no cycle in 462 neutrosophic graph.   (iv). By it's fixed-vertex and it's neutrosophic strong, all edges have same amount so 495 the connectedness amid two given edges is the same. All edges aren't type-IV. Since it's 496 impossible to define when there's no cycle in neutrosophic graph. 497 (v). By it's fixed-vertex and it's neutrosophic strong, all edges have same amount so 498 the connectedness amid two given edges is the same. All edges aren't type-V. Since it's 499 impossible to define when there's no cycle in neutrosophic graph. 500 (vi). By it's fixed-vertex and it's neutrosophic strong, all edges have same amount so 501 the connectedness amid two given edges is the same. All edges aren't type-VI. Since it's 502 impossible to define when there's no cycle in neutrosophic graph. 503 (vii). By it's fixed-vertex and it's neutrosophic strong, all edges have same amount 504 so the connectedness amid two given edges is the same. All edges aren't type-VII. Since 505 it's impossible to define when there's no cycle in neutrosophic graph. 526 (ii). All edges have same amount so the connectedness amid two given edges is the 527 same. All edges aren't type-II. By it's cycle, all vertices have 2 vertices which have 528 common edges which aren't type-II. Thus the set of representatives of colors is {}. The 529 type-II chromatic number is 0 and n-type-II chromatic number is 0. 530 (iii). All edges have same amount so the connectedness amid two given edges is the 531 same. All edges aren't type-III. By it's cycle, all vertices have 2 vertices which have 532 common edges which aren't type-III. Thus the set of representatives of colors is {}. The 533 type-III chromatic number is 0 and n-type-III chromatic number is 0. 534 (iv). All edges have same amount so the connectedness amid two given edges is the 535 same. All edges are type-IV. By it's cycle, all vertices have 2 vertices which have   (v). All edges have same amount so the connectedness amid two given edges is the 541 same. All edges aren't type-III. By it's cycle, all vertices have 2 vertices which have 542 common edges which aren't type-III. Thus the set of representatives of colors is {}. The 543 type-III chromatic number is 0 and n-type-III chromatic number is 0. 544 (vi). All edges have same amount so the connectedness amid two given edges is the 545 same. All edges aren't type-III. By it's cycle, all vertices have 2 vertices which have 546 common edges which aren't type-III. Thus the set of representatives of colors is {}. The 547 type-III chromatic number is 0 and n-type-III chromatic number is 0. 548 (vii). All edges have same amount so the connectedness amid two given edges is the 549 same. All edges aren't type-III. By it's cycle, all vertices have 2 vertices which have 550 common edges which aren't type-III. Thus the set of representatives of colors is {}. The 551 type-III chromatic number is 0 and n-type-III chromatic number is 0.  ii). All edges have same amount so the connectedness amid two given edges is the 574 same. All edges aren't type-II. By it's cycle, all vertices have 2 vertices which have 575 common edges which aren't type-II. Thus the set of representatives of colors is {}. The 576 type-II chromatic number is 0 and n-type-II chromatic number is 0. 577 (iii). All edges have same amount so the connectedness amid two given edges is the 578 same. All edges aren't type-III. By it's cycle, all vertices have 2 vertices which have 579 common edges which aren't type-III. Thus the set of representatives of colors is {}. The 580 type-III chromatic number is 0 and n-type-III chromatic number is 0. 581 (iv). All edges have same amount so the connectedness amid two given edges is the 582 same. All edges are type-IV. By it's cycle, all vertices have 2 vertices which have 583 common edges which are type-IV. By deletion of one edge, it's possible to compute 584 connectedness. Thus the set of representatives of colors is {v i , v j }. The type-IV 585 chromatic number is 2 and n-type-IV chromatic number is neutrosophic cardinality of (v). All edges have same amount so the connectedness amid two given edges is the 588 same. All edges aren't type-III. By it's cycle, all vertices have 2 vertices which have 589 common edges which aren't type-III. Thus the set of representatives of colors is {}. The 590 type-III chromatic number is 0 and n-type-III chromatic number is 0. 591 (vi). All edges have same amount so the connectedness amid two given edges is the 592 same. All edges aren't type-III. By it's cycle, all vertices have 2 vertices which have 593 common edges which aren't type-III. Thus the set of representatives of colors is {}. The 594 type-III chromatic number is 0 and n-type-III chromatic number is 0. 595 (vii). All edges have same amount so the connectedness amid two given edges is the 596 same. All edges aren't type-III. By it's cycle, all vertices have 2 vertices which have 597 common edges which aren't type-III. Thus the set of representatives of colors is {}. The 598 type-III chromatic number is 0 and n-type-III chromatic number is 0.  ii). All edges have same amount so the connectedness amid two given edges is the 621 same. All edges aren't type-II. By it's neutrosophic strong, there's a vertex has 2 622 vertices which have common edges which aren't type-II. Thus the set of representatives 623 of colors is {}. The type-II chromatic number is 0 and n-type-II chromatic number is 0. 624 (iii). All edges have same amount so the connectedness amid two given edges is the 625 same. All edges aren't type-III. By it's neutrosophic strong, there's a vertex has 2 626 vertices which have common edges which aren't type-III. Thus the set of representatives 627 of colors is {}. The type-III chromatic number is 0 and n-type-III chromatic number is 0. 628 (iv). All edges have same amount so the connectedness amid two given edges is the 629 same. All edges are type-IV. By it's cycle, all vertices have 2 vertices which have 630 common edges which are type-IV. Thus the set of representatives of colors is {v i , v j }.

631
The type-IV chromatic number is 2 and n-type-IV chromatic number is neutrosophic (v). All edges have same amount so the connectedness amid two given edges is the 634 same. All edges aren't type-V. By it's neutrosophic strong, there's a vertex has 2 635 vertices which have common edges which aren't type-V. Thus the set of representatives 636 of colors is {}. The type-V chromatic number is 0 and n-type-V chromatic number is 0. 637 (vi). All edges have same amount so the connectedness amid two given edges is the 638 same. All edges aren't type-VI. By it's cycle, all vertices have 2 vertices which have 639 common edges which aren't type-VI. Thus the set of representatives of colors is {}. The 640 type-VI chromatic number is 0 and n-type-VI chromatic number is 0. 641 (vii). All edges have same amount so the connectedness amid two given edges is the 642 same. All edges aren't type-VII. By it's cycle, all vertices have 2 vertices which have 643 common edges which aren't type-VII. Thus the set of representatives of colors is {}.

644
The type-VII chromatic number is 0 and n-type-VII chromatic number is 0.  ii). All edges have same amount so the connectedness amid two given edges is the 667 same. All edges aren't type-II. By it's neutrosophic strong, there's a vertex has 2 668 vertices which have common edges which aren't type-II. Thus the set of representatives 669 of colors is {}. The type-II chromatic number is 0 and n-type-II chromatic number is 0. 670 (iii). All edges have same amount so the connectedness amid two given edges is the 671 same. All edges aren't type-III. By it's neutrosophic strong, there's a vertex has 2 672 vertices which have common edges which aren't type-III. Thus the set of representatives 673 of colors is {}. The type-III chromatic number is 0 and n-type-III chromatic number is 0. 674 (iv). All edges have same amount so the connectedness amid two given edges is the 675 same. All edges are type-IV. By it's cycle, all vertices have 2 vertices which have 676 common edges which are type-IV. Thus the set of representatives of colors is {v i , v j }.

677
The type-IV chromatic number is 2 and n-type-IV chromatic number is neutrosophic (v). All edges have same amount so the connectedness amid two given edges is the 680 same. All edges aren't type-V. By it's neutrosophic strong, there's a vertex has 2 681 vertices which have common edges which aren't type-V. Thus the set of representatives 682 of colors is {}. The type-V chromatic number is 0 and n-type-V chromatic number is 0. 683 (vi). All edges have same amount so the connectedness amid two given edges is the 684 same. All edges aren't type-VI. By it's cycle, all vertices have 2 vertices which have 685 common edges which aren't type-VI. Thus the set of representatives of colors is {}. The 686 type-VI chromatic number is 0 and n-type-VI chromatic number is 0. 687 (vii). All edges have same amount so the connectedness amid two given edges is the 688 same. All edges aren't type-VII. By it's cycle, all vertices have 2 vertices which have 689 common edges which aren't type-VII. Thus the set of representatives of colors is {}.

892
Proof. (i). All edges have same amount so the connectedness amid two given edges is 893 the same. All edges are type-I. By it's neutrosophic complete, there's a vertex has t − 1 894 which have common edges which are type-I. Thus the set of representatives of colors is 895 {v 1 , v 2 , · · · , v t }. The type-I chromatic number is t and n-type-I chromatic number is 897 (ii). All edges have same amount so the connectedness amid two given edges is the 898 same. All edges aren't type-II. By it's neutrosophic complete, there's a vertex has t − 1 899 vertices which have common edges which aren't type-II. Thus the set of representatives 900 of colors is {}. The type-II chromatic number is 0 and n-type-II chromatic number is 0. 901 (iii). All edges have same amount so the connectedness amid two given edges is the 902 same. All edges aren't type-III. By it's neutrosophic complete, there's a vertex has t − 1 903 vertices which have common edges which aren't type-III. Thus the set of representatives 904 of colors is {}. The type-III chromatic number is 0 and n-type-III chromatic number is 0. 905 (iv). All edges have same amount so the connectedness amid two given edges is the 906 same. All edges are type-IV. By it's neutrosophic complete, there's a vertex has t − 1 907 vertices which have common edges which are type-IV. Thus the set of representatives of 908 colors is {v 1 , v 2 , · · · , v t }. The type-IV chromatic number is t and n-type-IV chromatic (v). All edges have same amount so the connectedness amid two given edges is the 912 same. All edges aren't type-V. By it's neutrosophic complete, there's a vertex has t − 1 913 vertices which have common edges which aren't type-V. Thus the set of representatives 914 of colors is {}. The type-V chromatic number is 0 and n-type-V chromatic number is 0. 915 (vi). All edges have same amount so the connectedness amid two given edges is the 916 same. All edges aren't type-VI. By it's neutrosophic complete, there's a vertex has t − 1 917 vertices which have common edges which aren't type-VI. Thus the set of representatives 918 of colors is {}. The type-VI chromatic number is 0 and n-type-VI chromatic number is 0. 919 (vii). All edges have same amount so the connectedness amid two given edges is the 920 same. All edges aren't type-VII. By it's neutrosophic complete, there's a vertex has number is t and n-type-I chromatic number is tσ (v i  and n-type-VII chromatic number is 0.

940
Proof. (i). All edges have same amount so the connectedness amid two given edges is 941 the same. All edges are type-I. By it's neutrosophic complete, there's a vertex has t − 1 942 vertices which have common edges which are type-I. Thus the set of representatives of 943 colors is {v 1 , v 2 , · · · , v t }. The type-I chromatic number is t and n-type-I chromatic 944 number is neutrosophic cardinality of {v 1 , v 2 , · · · , v t }. which is tσ(v i ). 945 (ii). All edges have same amount so the connectedness amid two given edges is the 946 same. All edges aren't type-II. By it's neutrosophic complete, there's a vertex has t − 1 947 vertices which have common edges which aren't type-II. Thus the set of representatives 948 of colors is {}. The type-II chromatic number is 0 and n-type-II chromatic number is 0. 949 (iii). All edges have same amount so the connectedness amid two given edges is the 950 same. All edges aren't type-III. By it's neutrosophic complete, there's a vertex has t − 1 951 vertices which have common edges which aren't type-III. Thus the set of representatives 952 of colors is {}. The type-III chromatic number is 0 and n-type-III chromatic number is 0. 953 (iv). All edges have same amount so the connectedness amid two given edges is the 954 same. All edges are type-IV. By it's neutrosophic complete, there's a vertex has t − 1 955 vertices which have common edges which are type-IV. Thus the set of representatives of 956 colors is {v 1 , v 2 , · · · , v t }. The type-IV chromatic number is t and n-type-IV chromatic 957 number is neutrosophic c{v 1 (v). All edges have same amount so the connectedness amid two given edges is the 959 same. All edges aren't type-V. By it's neutrosophic complete, there's a vertex has t − 1 960 vertices which have common edges which aren't type-V. Thus the set of representatives 961 of colors is {}. The type-V chromatic number is 0 and n-type-V chromatic number is 0. 962 (vi). All edges have same amount so the connectedness amid two given edges is the 963 same. All edges aren't type-VI. By it's neutrosophic complete, there's a vertex has t − 1 964 vertices which have common edges which aren't type-VI. Thus the set of representatives 965 of colors is {}. The type-VI chromatic number is 0 and n-type-VI chromatic number is 0. 966 (vii). All edges have same amount so the connectedness amid two given edges is the 967 same. All edges aren't type-VII. By it's neutrosophic complete, there's a vertex has 968 t − 1 vertices which have common edges which aren't type-VII. Thus the set of 969 representatives of colors is {}. The type-VII chromatic number is 0 and n-type-VII 970 chromatic number is 0.