ON FUZZY CHROMATIC NUMBER OF CARTESIAN PRODUCT OF SOME FUZZY GRAPHS AND ITS APPLICATION

2019 ◽  
Vol 20 (2) ◽  
pp. 237-252
Author(s):  
Isnaini Rosyida ◽  
Rosyida ◽  
Ch. Rini Indrati ◽  
Diari Indriati
Keyword(s):  
Chromatic Number ◽  
Cartesian Product ◽  
Fuzzy Graphs

On construction of fuzzy chromatic number of cartesian product of path and other fuzzy graphs

2020 ◽  
Vol 39 (1) ◽  
pp. 1073-1080
Author(s):  
Isnaini Rosyida ◽  
Widodo ◽  
Ch. Rini Indrati ◽  
Diari Indriati
Keyword(s):  
Chromatic Number ◽  
Cartesian Product ◽  
Fuzzy Graphs

scholarly journals Chromatic Number for a Generalization of Cartesian Product Graphs

10.37236/160 ◽  
2009 ◽  
Vol 16 (1) ◽  
Author(s):  
Daniel Král' ◽  
Douglas B. West
Keyword(s):  
Chromatic Number ◽  
Maximum Degree ◽  
Cartesian Product ◽  
Rectangular Grid ◽  
Product Graphs ◽  
Cartesian Product Graphs

Let ${\cal G}$ be a class of graphs. A $d$-fold grid over ${\cal G}$ is a graph obtained from a $d$-dimensional rectangular grid of vertices by placing a graph from ${\cal G}$ on each of the lines parallel to one of the axes. Thus each vertex belongs to $d$ of these subgraphs. The class of $d$-fold grids over ${\cal G}$ is denoted by ${\cal G}^d$. Let $f({\cal G};d)=\max_{G\in{\cal G}^d}\chi(G)$. If each graph in ${\cal G}$ is $k$-colorable, then $f({\cal G};d)\le k^d$. We show that this bound is best possible by proving that $f({\cal G};d)=k^d$ when ${\cal G}$ is the class of all $k$-colorable graphs. We also show that $f({\cal G};d)\ge{\left\lfloor\sqrt{{d\over 6\log d}}\right\rfloor}$ when ${\cal G}$ is the class of graphs with at most one edge, and $f({\cal G};d)\ge {\left\lfloor{d\over 6\log d}\right\rfloor}$ when ${\cal G}$ is the class of graphs with maximum degree $1$.

The b -chromatic number of the cartesian product of two graphs

2007 ◽  
Vol 44 (1) ◽  
pp. 49-55
Author(s):  
Mekkia Kouider ◽  
Maryvonne Mahéo
Keyword(s):  
Chromatic Number ◽  
Cartesian Product ◽  
Proper Coloring

In this paper we study the b -chromatic number of the cartesian product of two graphs. The b -chromatic number of a graph G is defined as the maximum number k of colors that can be used to color the vertices of G , such that we obtain a proper coloring and each color i has at least one representative χi adjacent to a vertex of every color j , 1 ≦ j ≠ i ≦ k . In this paper we get ρ( G□H ) ≦ ρ( G )( nH + 1) + Δ( H ) + 1, when the girth of G is assumed to be greater than or equal to 7.

scholarly journals Multiple Attribute Decision-Making Problem Using Picture Fuzzy Graph

2021 ◽  
Vol 2021 ◽  
pp. 1-16
Author(s):  
Sk. Amanathulla ◽  
G. Muhiuddin ◽  
D. Al-Kadi ◽  
M. Pal
Keyword(s):  
Decision Making ◽  
Cartesian Product ◽  
Multiple Attribute Decision Making ◽  
Fuzzy Environment ◽  
Fuzzy Graphs ◽  
Determination Process ◽  
Multiple Attribute ◽  
Picture Fuzzy Set ◽  
Almost All ◽  
The Relationship

In a picture fuzzy environment, almost all multiple attribute decision-making ( MADM ) methods have been discussed a type of problem in which there is no relationship among the attributes. Although the relationship among the attributes should be considered in the actual applications, so we need to pay attention to that important issue. This article applied graph theory to the picture fuzzy set ( PFS ) and obtained a new method, MADM , to solve complicated problems under a picture fuzzy environment. The developed method can capture the relationship among the attributes that cannot be handled well by any existing methods. This study introduces union, intersection, sum, Cartesian product, the composition of picture fuzzy graphs ( PFG s), and their important properties. Finally, by considering the importance of relationships among attributes in the determination process, two algorithms, based on PFG , have developed to solve complicated problems using picture fuzzy information. Also, two numerical examples have introduced to explain how to deal with the MADM problem under picture fuzzy environment.

scholarly journals BILANGAN KROMATIK LOKASI DARI GRAF P m P n ; K m P n ; DAN K , m K n

2013 ◽  
Vol 2 (1) ◽  
pp. 14
Author(s):  
Mariza Wenni
Keyword(s):  
Natural Number ◽  
Chromatic Number ◽  
Cartesian Product ◽  
Complete Graphs ◽  
Chromatic Numbers ◽  
Color Code ◽  
Connected Graphs ◽  
Vertex Set ◽  
Distinct Color

Let G and H be two connected graphs. Let c be a vertex k-coloring of aconnected graph G and let = fCg be a partition of V (G) into the resultingcolor classes. For each v 2 V (G), the color code of v is dened to be k-vector: c1; C2; :::; Ck(v) =(d(v; C1); d(v; C2); :::; d(v; Ck)), where d(v; Ci) = minfd(v; x) j x 2 Cg, 1 i k. Ifdistinct vertices have distinct color codes with respect to , then c is called a locatingcoloring of G. The locating chromatic number of G is the smallest natural number ksuch that there are locating coloring with k colors in G. The Cartesian product of graphG and H is a graph with vertex set V (G) V (H), where two vertices (a; b) and (a)are adjacent whenever a = a0and bb02 E(H), or aa0i2 E(G) and b = b, denotedby GH. In this paper, we will study about the locating chromatic numbers of thecartesian product of two paths, the cartesian product of paths and complete graphs, andthe cartesian product of two complete graphs.

scholarly journals CHROMATIC NUMBER OF BIPOLAR FUZZY GRAPHS

2016 ◽  
Vol 34 (1_2) ◽  
pp. 49-60 ◽  
Author(s):  
A. TAHMASBPOUR ◽  
R.A. BORZOOEI
Keyword(s):  
Chromatic Number ◽  
Fuzzy Graphs ◽  
Bipolar Fuzzy Graphs

Star chromatic number of some graphs

2021 ◽  
pp. 2150089
Author(s):  
S. Akbari ◽  
M. CHAVOOSHI ◽  
M. Ghanbari ◽  
S. Taghian
Keyword(s):  
Chromatic Number ◽  
Cartesian Product ◽  
Vertex Coloring ◽  
Star Coloring ◽  
Proper Vertex

A proper vertex coloring of a graph [Formula: see text] is called a star coloring if every two color classes induce a forest whose each component is a star, which means there is no bicolored [Formula: see text] in [Formula: see text]. In this paper, we show that the Cartesian product of any two cycles, except [Formula: see text] and [Formula: see text], has a [Formula: see text]-star coloring.

scholarly journals k -tuple chromatic number of the cartesian product of graphs

2015 ◽  
Vol 50 ◽  
pp. 243-248
Author(s):  
Flavia Bonomo ◽  
Ivo Koch ◽  
Pablo Torres ◽  
Mario Valencia-Pabon
Keyword(s):  
Chromatic Number ◽  
Cartesian Product ◽  
Cartesian Product Of Graphs ◽  
Product Of Graphs
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