scholarly journals Chromatic Number and Neutrosophic Chromatic Number

2021 ◽  
Author(s):  
Henry Garrett
Keyword(s):  
Graph Theory ◽  
Chromatic Number ◽  
Common Edge ◽  
Specific Relation ◽  
Neutrosophic Graph

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.

scholarly journals Neutrosophic Chromatic Number Based on Connectedness

2021 ◽  
Author(s):  
Henry Garrett
Keyword(s):  
Graph Theory ◽  
Chromatic Number ◽  
Common Edge ◽  
Specific Relation ◽  
Neutrosophic Graph

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.

scholarly journals Different Types of Neutrosophic Chromatic Number

2021 ◽  
Author(s):  
Henry Garrett
Keyword(s):  
Graph Theory ◽  
Chromatic Number ◽  
Chromatic Numbers ◽  
Common Edge ◽  
Specific Relation ◽  
Different Types ◽  
Neutrosophic Graph

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.

scholarly journals Metric Dimension in fuzzy(neutrosophic) Graphs-VII

2021 ◽  
Author(s):  
Henry Garrett
Keyword(s):  
Graph Theory ◽  
Individual Case ◽  
Metric Dimension ◽  
Neutrosophic Graph ◽  
Distinct Distance ◽  
Optimal Set ◽  
Dimension Number

New notion of dimension as set, as two optimal numbers including metric number, dimension number and as optimal set are introduced in individual framework and in formation of family. Behaviors of twin and antipodal are explored in fuzzy(neutrosophic) graphs. Fuzzy(neutrosophic) graphs, under conditions, fixed-edges, fixed-vertex and strong fixed-vertex are studied. Some classes as path, cycle, complete, strong, t-partite, bipartite, star and wheel in the formation of individual case and in the case, they form a family are studied in the term of dimension. Fuzzification(neutrosofication) of twin vertices but using crisp concept of antipodal vertices are another approaches of this study. Thus defining two notions concerning vertices which one of them is fuzzy(neutrosophic) titled twin and another is crisp titled antipodal to study the behaviors of cycles which are partitioned into even and odd, are concluded. Classes of cycles according to antipodal vertices are divided into two classes as even and odd. Parity of the number of edges in cycle causes to have two subsections under the section is entitled to antipodal vertices. In this study, the term dimension is introduced on fuzzy(neutrosophic) graphs. The locations of objects by a set of some junctions which have distinct distance from any couple of objects out of the set, are determined. Thus it’s possible to have the locations of objects outside of this set by assigning partial number to any objects. The classes of these specific graphs are chosen to obtain some results based on dimension. The types of crisp notions and fuzzy(neutrosophic) notions are used to make sense about the material of this study and the outline of this study uses some new notions which are crisp and fuzzy(neutrosophic). Some questions and problems are posed concerning ways to do further studies on this topic. Basic familiarities with fuzzy(neutrosophic) graph theory and graph theory are proposed for this article.

Chromatic-durable graphs

1972 ◽  
Vol 71 (2) ◽  
pp. 227-233
Author(s):  
Don R. Lick ◽  
Arthur T. White
Keyword(s):  
Graph Theory ◽  
Chromatic Number ◽  
Intimate Relationship ◽  
Colour Problem

One of the most studied parameters in all of graph theory is the chromatic number. Undoubtedly, its popularity as a subject for research is due to its intimate relationship with the famous Four Colour Problem.

scholarly journals A Note on Embedding Hypertrees

10.37236/256 ◽  
2009 ◽  
Vol 16 (1) ◽  
Author(s):  
Po-Shen Loh
Keyword(s):  
Graph Theory ◽  
Chromatic Number ◽  
Classical Result ◽  
Uniform Hypergraph ◽  
Uniform Hypergraphs

A classical result from graph theory is that every graph with chromatic number $\chi > t$ contains a subgraph with all degrees at least $t$, and therefore contains a copy of every $t$-edge tree. Bohman, Frieze, and Mubayi recently posed this problem for $r$-uniform hypergraphs. An $r$-tree is a connected $r$-uniform hypergraph with no pair of edges intersecting in more than one vertex, and no sequence of distinct vertices and edges $(v_1, e_1, \ldots, v_k, e_k)$ with all $e_i \ni \{v_i, v_{i+1}\}$, where we take $v_{k+1}$ to be $v_1$. Bohman, Frieze, and Mubayi proved that $\chi > 2rt$ is sufficient to embed every $r$-tree with $t$ edges, and asked whether the dependence on $r$ was necessary. In this note, we completely solve their problem, proving the tight result that $\chi > t$ is sufficient to embed any $r$-tree with $t$ edges.

scholarly journals Region Coloring in Minahasa Regency Using Sequential Color Algorithm

d'CARTESIAN ◽  
2019 ◽  
Vol 8 (2) ◽  
pp. 86
Author(s):  
Yevie Ingamita ◽  
Nelson Nainggolan ◽  
Benny Pinontoan
Keyword(s):  
Graph Theory ◽  
Positive Integer ◽  
Chromatic Number ◽  
Human Life ◽  
Minimum Number ◽  
Mathematical Sciences ◽  
Map Coloring ◽  
Graph Application

Graph Theory is one of the mathematical sciences whose application is very wide in human life. One of theory graph application is Map Coloring. This research discusses how to color the map of Minahasa Regency by using the minimum color that possible. The algorithm used to determine the minimum color in coloring the region of Minahasa Regency that is Sequential Color Algorithm. The Sequential Color Algorithm is an algorithm used in coloring a graph with k-color, where k is a positive integer. Based on the results of this research was found that the Sequential Color Algorithm can be used to color the map of Minahasa Regency with the minimum number of colors or chromatic number χ(G) obtained in the coloring of 25 sub-districts on the map of Minahasa Regency are 3 colors (χ(G) = 3).

scholarly journals Graphical Analysis of Covering and Paired Domination in the Environment of Neutrosophic Information

2021 ◽  
Vol 2021 ◽  
pp. 1-12
Author(s):  
Sami Ullah Khan ◽  
Abdul Nasir ◽  
Naeem Jan ◽  
Zhen-Hua Ma
Keyword(s):  
Graph Theory ◽  
Optimization Problems ◽  
Domination Number ◽  
Real Life ◽  
Graphical Analysis ◽  
Fuzzy Graphs ◽  
Life Problems ◽  
Neutrosophic Graph ◽  
Paired Domination ◽  
Intuitionistic Fuzzy Graphs

Neutrosophic graph (NG) is a powerful tool in graph theory, which is capable of modeling many real-life problems with uncertainty due to unclear, varying, and indeterminate information. Meanwhile, the fuzzy graphs (FGs) and intuitionistic fuzzy graphs (IFGs) may not handle these problems as efficiently as NGs. It is difficult to model uncertainty due to imprecise information and vagueness in real-world scenarios. Many real-life optimization problems are modeled and solved using the well-known fuzzy graph theory. The concepts of covering, matching, and paired domination play a major role in theoretical and applied neutrosophic environments of graph theory. Henceforth, the current study covers this void by introducing the notions of covering, matching, and paired domination in single-valued neutrosophic graph (SVNG) using the strong edges. Also, many attention-grabbing properties of these concepts are studied. Moreover, the strong covering number, strong matching number, and the strong paired domination number of complete SVNG, complete single-valued neutrosophic cycle (SVNC), and complete bipartite SVNG are worked out along with their fascinating properties.

BEBERAPA SYARAT CUKUP UNTUK BILANGAN KROMATIK LOKASI HINGGA PADA GRAF TAK TERHUBUNG

2018 ◽  
Vol 19 (1) ◽  
pp. 76-82
Author(s):  
Des Welyyanti
Keyword(s):  
Graph Theory ◽  
Chromatic Number ◽  
Double Stars ◽  
Disconnected Graphs

The locating-chromatic number of a graph is introduced by Chartrand et al. in 2002. Firstly, Chatrand et al. determine the locating-chromatic number of path and double stars. The locating-chromatic number is an interesting concept ini graph theory. In this paper, we determine some condtions for  disconnected graphs has a finite locating-chromatic number.

scholarly journals A Study of Regular and Irregular Neutrosophic Graphs with Real Life Applications

Mathematics ◽  
2019 ◽  
Vol 7 (6) ◽  
pp. 551 ◽  
Author(s):  
Liangsong Huang ◽  
Yu Hu ◽  
Yuxia Li ◽  
P. K. Kishore Kumar ◽  
Dipak Koley ◽  
...  
Keyword(s):  
Graph Theory ◽  
Optimization Problems ◽  
Real Life ◽  
Road Transport ◽  
Fuzzy Graph ◽  
Fuzzy Graphs ◽  
Life Problems ◽  
Real World Problem ◽  
Neutrosophic Graph ◽  
Definition Of

Fuzzy graph theory is a useful and well-known tool to model and solve many real-life optimization problems. Since real-life problems are often uncertain due to inconsistent and indeterminate information, it is very hard for an expert to model those problems using a fuzzy graph. A neutrosophic graph can deal with the uncertainty associated with the inconsistent and indeterminate information of any real-world problem, where fuzzy graphs may fail to reveal satisfactory results. The concepts of the regularity and degree of a node play a significant role in both the theory and application of graph theory in the neutrosophic environment. In this work, we describe the utility of the regular neutrosophic graph and bipartite neutrosophic graph to model an assignment problem, a road transport network, and a social network. For this purpose, we introduce the definitions of the regular neutrosophic graph, star neutrosophic graph, regular complete neutrosophic graph, complete bipartite neutrosophic graph, and regular strong neutrosophic graph. We define the d m - and t d m -degrees of a node in a regular neutrosophic graph. Depending on the degree of the node, this paper classifies the regularity of a neutrosophic graph into three types, namely d m -regular, t d m -regular, and m-highly irregular neutrosophic graphs. We present some theorems and properties of those regular neutrosophic graphs. The concept of an m-highly irregular neutrosophic graph on cycle and path graphs is also investigated in this paper. The definition of busy and free nodes in a regular neutrosophic graph is presented here. We introduce the idea of the μ -complement and h-morphism of a regular neutrosophic graph. Some properties of complement and isomorphic regular neutrosophic graphs are presented here.

scholarly journals On Uniquely 3-Colorable Plane Graphs without Adjacent Faces of Prescribed Degrees

Mathematics ◽  
2019 ◽  
Vol 7 (9) ◽  
pp. 793
Author(s):  
Zepeng Li ◽  
Naoki Matsumoto ◽  
Enqiang Zhu ◽  
Jin Xu ◽  
Tommy Jensen
Keyword(s):  
Chromatic Number ◽  
Infinite Family ◽  
Plane Graph ◽  
Common Edge ◽  
Plane Graphs

A graph G is uniquely k-colorable if the chromatic number of G is k and G has only one k-coloring up to the permutation of the colors. For a plane graph G, two faces f 1 and f 2 of G are adjacent ( i , j )-faces if d ( f 1 ) = i, d ( f 2 ) = j, and f 1 and f 2 have a common edge, where d ( f ) is the degree of a face f. In this paper, we prove that every uniquely three-colorable plane graph has adjacent ( 3 , k )-faces, where k ≤ 5. The bound of five for k is the best possible. Furthermore, we prove that there exists a class of uniquely three-colorable plane graphs having neither adjacent ( 3 , i )-faces nor adjacent ( 3 , j )-faces, where i , j are fixed in { 3 , 4 , 5 } and i ≠ j. One of our constructions implies that there exists an infinite family of edge-critical uniquely three-colorable plane graphs with n vertices and 7 3 n - 14 3 edges, where n ( ≥ 11 ) is odd and n ≡ 2 ( mod 3 ).

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