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 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 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.

scholarly journals On Graphs Related to Comaximal Ideals of a Commutative Ring

2013 ◽  
Vol 2013 ◽  
pp. 1-7 ◽  
Author(s):  
Tongsuo Wu ◽  
Meng Ye ◽  
Dancheng Lu ◽  
Houyi Yu
Keyword(s):  
Commutative Ring ◽  
Chromatic Number ◽  
Clique Number ◽  
Induced Subgraph ◽  
The Core ◽  
Maximal Ideals ◽  
Graph Properties ◽  
Vertex Set

We study the co maximal graph Ω(R), the induced subgraph Γ(R) of Ω(R) whose vertex set is R∖(U(R)∪J(R)), and a retract Γr(R) of Γ(R), where R is a commutative ring. For a graph Γ(R) which contains a cycle, we show that the core of Γ(R) is a union of triangles and rectangles, while a vertex in Γ(R) is either an end vertex or a vertex in the core. For a nonlocal ring R, we prove that both the chromatic number and clique number of Γ(R) are identical with the number of maximal ideals of R. A graph Γr(R) is also introduced on the vertex set {Rx∣x∈R∖(U(R)∪J(R))}, and graph properties of Γr(R) are studied.

ON GRAPHS ASSOCIATED TO CONJUGACY CLASSES OF SOME THREE-GENERATOR GROUPS

2016 ◽  
Vol 79 (1) ◽  
Author(s):  
Nor Haniza Sarmin ◽  
Alia Husna Mohd Noor ◽  
Sanaa Mohamed Saleh Omer
Keyword(s):  
Conjugacy Class ◽  
Chromatic Number ◽  
Clique Number ◽  
Conjugacy Classes ◽  
Line Segments ◽  
Dominating Number ◽  
Commuting Graph ◽  
Independent Number ◽  
Graph Properties ◽  
Class Graph

A graph consists of points which are called vertices, and connections which are called edges, which are indicated by line segments or curves joining certain pairs of vertices.  In this paper, four types of graphs which are the commuting graph, non-commuting graph conjugate graph and the conjugacy class graph for some three-generator groups are discussed. Some of the graph properties are also found which include the independent number, chromatic number, clique number and dominating number.

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.

scholarly journals Locating-chromatic number of the edge-amalgamation of trees

2020 ◽  
Vol 4 (2) ◽  
pp. 126
Author(s):  
Dian Kastika Syofyan ◽  
Edy Tri Baskoro ◽  
Hilda Assiyatun
Keyword(s):  
Chromatic Number ◽  
Upper Bounds ◽  
Title Page ◽  
Metric Dimension ◽  
Chromatic Numbers ◽  
Lower And Upper Bounds ◽  
Vertex Set ◽  
Partition Dimension ◽  
Special Case

<div class="page" title="Page 1"><div class="layoutArea"><div class="column"><p><span>The investigation on the locating-chromatic number of a graph was initiated by Chartrand </span><span>et al. </span><span>(2002). This concept is in fact a special case of the partition dimension of a graph. This topic has received much attention. However, the results are still far from satisfaction. We can define the locating-chromatic number of a graph </span><span>G </span><span>as the smallest integer </span><span>k </span><span>such that there exists a </span><span>k</span><span>-partition of the vertex-set of </span><span>G </span><span>such that all vertices have distinct coordinates with respect to this partition. As we know that the metric dimension of a tree is completely solved. However, the locating-chromatic numbers for most of trees are still open. For </span><span><em>i</em> </span><span>= 1</span><span>, </span><span>2</span><span>, . . . , <em>t</em>, </span><span>let </span><em>T</em><span>i </span><span>be a tree with a fixed edge </span><span>e</span><span>o</span><span>i </span><span>called the terminal edge. The edge-amalgamation of all </span><span>T</span><span>i</span><span>s </span><span>denoted by Edge-Amal</span><span>{</span><span>T</span><span>i</span><span>;</span><span>e</span><span>o</span><span>i</span><span>} </span><span>is a tree formed by taking all the </span><span>T</span><span>i</span><span>s and identifying their terminal edges. In this paper, we study the locating-chromatic number of the edge-amalgamation of arbitrary trees. We give lower and upper bounds for their locating-chromatic numbers and show that the bounds are tight.</span></p></div></div></div>

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