scholarly journals On the Locating Chromatic Number of Barbell Shadow Path Graph

2021 ◽  
Vol 5 (2) ◽  
pp. 82
Author(s):  
A. Asmiati ◽  
Maharani Damayanti ◽  
Lyra Yulianti
Keyword(s):  
Chromatic Number ◽  
Path Graph ◽  
Partition Dimension

The locating-chromatic number was introduced by Chartrand in 2002. The locating chromatic number of a graph is a combined concept between the coloring and partition dimension of a graph. The locating chromatic number of a graph is defined as the cardinality of the minimum color classes of the graph. In this paper, we discuss about the locating-chromatic number of shadow path graph and barbell graph containing shadow graph.

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 The Locating-Chromatic Number of Origami Graphs

Algorithms ◽  
2021 ◽  
Vol 14 (6) ◽  
pp. 167
Author(s):  
Agus Irawan ◽  
Asmiati Asmiati ◽  
La Zakaria ◽  
Kurnia Muludi
Keyword(s):  
Chromatic Number ◽  
Outer Edge ◽  
Partition Dimension

The locating-chromatic number of a graph combines two graph concepts, namely coloring vertices and partition dimension of a graph. The locating-chromatic number is the smallest k such that G has a locating k-coloring, denoted by χL(G). This article proposes a procedure for obtaining a locating-chromatic number for an origami graph and its subdivision (one vertex on an outer edge) through two theorems with proofs.

scholarly journals The Partition Dimension of a Path Graph

2021 ◽  
Vol 13 (2) ◽  
pp. 66
Author(s):  
Vivi Ramdhani ◽  
Fathur Rahmi
Keyword(s):  
Graph Theory ◽  
Connected Graph ◽  
Ordered Set ◽  
Minimum Cardinality ◽  
Path Graph ◽  
Partition Dimension

Resolving partition is part of graph theory. This article, explains about resolving partition of the path graph, with. Given a connected graph  and  is a subset of  writen . Suppose there is , then the distance between and  is denoted in the form . There is an ordered set of -partitions of, writen then  the representation of with respect tois the  The set of partitions ofis called a resolving partition if the representation of each  to  is different. The minimum cardinality of the solving-partition to  is called the partition dimension of G which is denoted by . Before getting the partition dimension of a path graph, the first step is to look for resolving partition of the graph. Some resolving partitions of path graph,  with ,  and  are obtained. Then, the partition dimension of the path graph which is the minimum cardinality of resolving partition, namely pd (Pn)=2Resolving partition is part of graph theory. This article, explains about resolving partition of the path graph, with. Given a connected graph  and  is a subset of  writen . Suppose there is , then the distance between and  is denoted in the form . There is an ordered set of -partitions of, writen then  the representation of with respect tois the  The set of partitions ofis called a resolving partition if the representation of each  to  is different. The minimum cardinality of the solving-partition to  is called the partition dimension of G which is denoted by . Before getting the partition dimension of a path graph, the first step is to look for resolving partition of the graph. Some resolving partitionsof path graph, with ,  and  are obtained. Then, the partition dimension of the path graph which is the minimum cardinality of resolving partition, namely.

scholarly journals ON STAR COLORING OF DEGREE SPLITTING OF COMB PRODUCT GRAPHS

2020 ◽  
pp. 507
Author(s):  
Ulagammal Subramanian ◽  
Vernold Vivin Joseph
Keyword(s):  
Chromatic Number ◽  
Complete Graph ◽  
Vertex Coloring ◽  
Star Graph ◽  
Path Graph ◽  
Product Graphs ◽  
Star Coloring ◽  
Splitting Graph ◽  
Proper Vertex

A star coloring of a graph G is a proper vertex coloring in which every path on four vertices in G is not bicolored. The star chromatic number χs (G) of G is the least number of colors needed to star color G. Let G = (V,E) be a graph with V = S1 [ S2 [ S3 [ . . . [ St [ T where each Si is a set of all vertices of the same degree with at least two elements and T =V (G) − St i=1 Si. The degree splitting graph DS (G) is obtained by adding vertices w1,w2, . . .wt and joining wi to each vertex of Si for 1 i t. The comb product between two graphs G and H, denoted by G ⊲ H, is a graph obtained by taking one copy of G and |V (G)| copies of H and grafting the ith copy of H at the vertex o to the ith vertex of G. In this paper, we give the exact value of star chromatic number of degree splitting of comb product of complete graph with complete graph, complete graph with path, complete graph with cycle, complete graph with star graph, cycle with complete graph, path with complete graph and cycle with path graph.

scholarly journals On r-dynamic chromatic number of coronation of order two of any graphs with path graph

2019 ◽  
Vol 243 ◽  
pp. 012113
Author(s):  
B J Septory ◽  
Dafik ◽  
A I Kristiana ◽  
I H Agustin ◽  
D A R Wardani
Keyword(s):  
Chromatic Number ◽  
Path Graph

scholarly journals The locating-chromatic number and partition dimension of fibonacene graphs

2020 ◽  
Vol 3 (2) ◽  
pp. 116 ◽  
Author(s):  
Ratih Suryaningsih ◽  
Edy Tri Baskoro
Keyword(s):  
Chromatic Number ◽  
Fibonacci Numbers ◽  
Benzenoid Hydrocarbons ◽  
Graph Properties ◽  
Partition Dimension ◽  
Independence Numbers

<p>Fibonacenes are unbranched catacondensed benzenoid hydrocarbons in which all the non-terminal hexagons are angularly annelated. A hexagon is said to be angularly annelated if the hexagon is adjacent to exactly two other hexagons and possesses two adjacent vertices of degree 2. Fibonacenes possess remarkable properties related with Fibonacci numbers. Various graph properties of fibonacenes have been extensively studied, such as their saturation numbers, independence numbers and Wiener indices. In this paper, we show that the locating-chromatic number of any fibonacene graph is 4 and the partition dimension of such a graph is 3.</p>

scholarly journals b-CHROMATIC NUMBER OF CORONA PRODUCT OF CROWN GRAPH AND COMPLETE BIPARTITE GRAPH WITH PATH GRAPH

2015 ◽  
Vol 2 (2) ◽  
pp. 30-33
Author(s):  
Vijayalakshmi D ◽  
Mohanappriya G
Keyword(s):  
Bipartite Graph ◽  
Structural Properties ◽  
Chromatic Number ◽  
Complete Bipartite Graph ◽  
Proper Coloring ◽  
Path Graph ◽  
Crown Graph ◽  
Corona Product

A b-coloring of a graph is a proper coloring where each color admits at least one node (called dominating node) adjacent to every other used color. The maximum number of colors needed to b-color a graph G is called the b-chromatic number and is denoted by φ(G). In this paper, we find the b-chromatic number and some of the structural properties of corona product of crown graph and complete bipartite graphwith path graph.

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.

scholarly journals On the Locating Chromatic Number of Certain Barbell Graphs

2018 ◽  
Vol 2018 ◽  
pp. 1-5
Author(s):  
Asmiati ◽  
I. Ketut Sadha Gunce Yana ◽  
Lyra Yulianti
Keyword(s):  
Chromatic Number ◽  
Graph Partition ◽  
Vertex Set ◽  
Partition Dimension ◽  
Partition Class ◽  
Special Case

The locating chromatic number of a graph G is defined as the cardinality of a minimum resolving partition of the vertex set V(G) such that all vertices have distinct coordinates with respect to this partition and every two adjacent vertices in G are not contained in the same partition class. In this case, the coordinate of a vertex v in G is expressed in terms of the distances of v to all partition classes. This concept is a special case of the graph partition dimension notion. In this paper we investigate the locating chromatic number for two families of barbell graphs.

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