scholarly journals Determination of Fractional Chromatic Numbers in the Operation of Adding Two Different Graphs

2022 ◽  
Vol 18 (2) ◽  
pp. 161-168
Author(s):  
Junianto Sesa ◽  
Siswanto Siswanto
Keyword(s):  
Graph Theory ◽  
Chromatic Number ◽  
Index Theory ◽  
Chromatic Numbers ◽  
Fractional Chromatic Number ◽  
Path Graph ◽  
Fractional Coloring ◽  
Different Color ◽  
Cycle Graph

The development of graph theory has provided many new pieces of knowledge, one of them is graph color. Where the application is spread in various fields such as the coding index theory. Fractional coloring is multiple coloring at points with different colors where the adjoining point has a different color. The operation in the graph is known as the sum operation. Point coloring can be applied to graphs where the result of operations is from several special graphs.  In this case, the graph summation results of the path graph and the cycle graph will produce the same fractional chromatic number as the sum of the fractional chromatic numbers of each graph before it is operated.

Packing coloring on subdivision-vertex and subdivision-edge join of cycle Cm with path Pn

2020 ◽  
pp. 627-634
Author(s):  
K. Rajalakshmi ◽  
M. Venkatachalam ◽  
M. Barani ◽  
D. Dafik
Keyword(s):  
Chromatic Number ◽  
Path Graph ◽  
Packing Chromatic Number ◽  
Cycle Graph

The packing chromatic number $\chi_\rho$ of a graph $G$ is the smallest integer $k$ for which there exists a mapping $\pi$ from $V(G)$ to $\{1,2,...,k\}$ such that any two vertices of color $i$ are at distance at least $i+1$. In this paper, the authors find the packing chromatic number of subdivision vertex join of cycle graph with path graph and subdivision edge join of cycle graph with path graph.

scholarly journals Fractional Coloring of Triangle-Free Planar Graphs

10.37236/4135 ◽  
2015 ◽  
Vol 22 (4) ◽  
Author(s):  
Zdeněk Dvořák ◽  
Jean-Sébastien Sereni ◽  
Jan Volec
Keyword(s):  
Chromatic Number ◽  
Planar Graphs ◽  
Free Graph ◽  
Fractional Chromatic Number ◽  
Fractional Coloring

We prove that every planar triangle-free graph on $n$ vertices has fractional chromatic number at most $3-3/(3n+1)$.

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 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 Cubical coloring — fractional covering by cuts and semidefinite programming

2015 ◽  
Vol Vol. 17 no.2 (Graph Theory) ◽  
Author(s):  
Robert Šámal
Keyword(s):  
Semidefinite Programming ◽  
Chromatic Number ◽  
Fractional Chromatic Number ◽  
Maximum Cut ◽  
Continuous Mappings ◽  
Polynomial Time Approximation Algorithm ◽  
Eigenvalue Bound ◽  
International Audience ◽  
Graph Parameters ◽  
Fractional Covering

International audience We introduce a new graph parameter that measures fractional covering of a graph by cuts. Besides being interesting in its own right, it is useful for study of homomorphisms and tension-continuous mappings. We study the relations with chromatic number, bipartite density, and other graph parameters. We find the value of our parameter for a family of graphs based on hypercubes. These graphs play for our parameter the role that cliques play for the chromatic number and Kneser graphs for the fractional chromatic number. The fact that the defined parameter attains on these graphs the correct value suggests that our definition is a natural one. In the proof we use the eigenvalue bound for maximum cut and a recent result of Engström, Färnqvist, Jonsson, and Thapper [An approximability-related parameter on graphs – properties and applications, DMTCS vol. 17:1, 2015, 33–66]. We also provide a polynomial time approximation algorithm based on semidefinite programming and in particular on vector chromatic number (defined by Karger, Motwani and Sudan [Approximate graph coloring by semidefinite programming, J. ACM 45 (1998), no. 2, 246–265]).

scholarly journals On r-dynamic coloring of comb graphs

2021 ◽  
Vol 27 (2) ◽  
pp. 191-200
Author(s):  
K. Kalaiselvi ◽  
◽  
N. Mohanapriya ◽  
J. Vernold Vivin ◽  
◽  
...  
Keyword(s):  
Lower Bound ◽  
Chromatic Number ◽  
Upper Bound ◽  
Line Graph ◽  
Proper Coloring ◽  
Total Graph ◽  
Middle Graph ◽  
Different Color

An r-dynamic coloring of a graph G is a proper coloring of G such that every vertex in V(G) has neighbors in at least $\min\{d(v),r\}$ different color classes. The r-dynamic chromatic number of graph G denoted as $\chi_r (G)$, is the least k such that G has a coloring. In this paper we obtain the r-dynamic chromatic number of the central graph, middle graph, total graph, line graph, para-line graph and sub-division graph of the comb graph $P_n\odot K_1$ denoted by $C(P_n\odot K_1), M(P_n\odot K_1), T(P_n\odot K_1), L(P_n\odot K_1), P(P_n\odot K_1)$ and $S(P_n\odot K_1)$ respectively by finding the upper bound and lower bound for the r-dynamic chromatic number of the Comb graph.

scholarly journals Dimensi Metrik Kuat Lokal Graf Hasil Operasi Kali Kartesian

2020 ◽  
Vol 1 (2) ◽  
pp. 64
Author(s):  
Nurma Ariska Sutardji ◽  
Liliek Susilowati ◽  
Utami Dyah Purwati
Keyword(s):  
Graph Theory ◽  
Cartesian Product ◽  
Metric Dimension ◽  
Star Graph ◽  
Product Graph ◽  
Path Graph ◽  
Strong Metric Dimension ◽  
Metric Dimensions ◽  
Development Result ◽  
Corona Product

The strong local metric dimension is the development result of a strong metric dimension study, one of the study topics in graph theory. Some of graphs that have been discovered about strong local metric dimension are path graph, star graph, complete graph, cycle graphs, and the result corona product graph. In the previous study have been built about strong local metric dimensions of corona product graph. The purpose of this research is to determine the strong local metric dimension of cartesian product graph between any connected graph G and H, denoted by dimsl (G x H). In this research, local metric dimension of G x H is influenced by local strong metric dimension of graph G and local strong metric dimension of graph H. Graph G and graph H has at least two order.

The 3D graph approach for breakdown voltage calculation in BaTiO3 ceramics

2021 ◽  
pp. 2150103
Author(s):  
Vojislav V. Mitic ◽  
Branislav Randjelovic ◽  
Ivana Ilic ◽  
Srdjan Ribar ◽  
An-Lu Chun ◽  
...  
Keyword(s):  
Graph Theory ◽  
Crystal Structures ◽  
Breakdown Voltage ◽  
Main Idea ◽  
Experimental Results ◽  
Grain Structure ◽  
Mathematical Approach ◽  
Electronic Parameters ◽  
Mathematical Graph

After pioneering attempts for the introduction of graph theory in the field of ceramics and microstructures, where 1D and 2D graphs were used, in this paper we applied 3D graphs for the breakdown voltage calculation in BaTiO3 sample with some predefined constraints. We have described the relations between grains in the sample and established a mathematical approach for the calculation of breakdown voltage using experimental results. As a result, we introduced mapping between the property of sample and grain structure, then between the grain structure and mathematical graph, using various crystal structures. The main idea was to apply 3D graph theory for the distribution of electronic parameters between the neighboring grains. With this study, we successfully confirmed the possibilities for applications of graphs as a tool for the determination of properties even at the intergranular level.

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