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>

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.

The Weakly Nilpotent Graph of a Commutative Ring

2017 ◽  
Vol 60 (2) ◽  
pp. 319-328
Author(s):  
Soheila Khojasteh ◽  
Mohammad Javad Nikmehr
Keyword(s):  
Commutative Ring ◽  
Chromatic Number ◽  
Disjoint Union ◽  
Independence Number ◽  
Artinian Ring ◽  
Clique Number ◽  
Commutative Rings ◽  
Unicyclic Graph ◽  
Nilpotent Elements ◽  
Vertex Set

AbstractLet R be a commutative ring with non-zero identity. In this paper, we introduce theweakly nilpotent graph of a commutative ring. The weakly nilpotent graph of R denoted by Γw(R) is a graph with the vertex set R* and two vertices x and y are adjacent if and only if x y ∊ N(R)*, where R* = R \ {0} and N(R)* is the set of all non-zero nilpotent elements of R. In this article, we determine the diameter of weakly nilpotent graph of an Artinian ring. We prove that if Γw(R) is a forest, then Γw(R) is a union of a star and some isolated vertices. We study the clique number, the chromatic number, and the independence number of Γw(R). Among other results, we show that for an Artinian ring R, Γw(R) is not a disjoint union of cycles or a unicyclic graph. For Artinan rings, we determine diam . Finally, we characterize all commutative rings R for which is a cycle, where is the complement of the weakly nilpotent graph of R.

THE JACOBSON GRAPH OF COMMUTATIVE RINGS

2012 ◽  
Vol 12 (03) ◽  
pp. 1250179 ◽  
Author(s):  
A. AZIMI ◽  
A. ERFANIAN ◽  
M. FARROKHI D. G.
Keyword(s):  
Commutative Ring ◽  
Chromatic Number ◽  
Independence Number ◽  
Commutative Rings ◽  
Dominating Number ◽  
Jacobson Graph ◽  
Vertex Set ◽  
Numerical Invariants ◽  
Nonzero Identity

Let R be a commutative ring with nonzero identity. Then the Jacobson graph of R, denoted by 𝔍R, is defined as a graph with vertex set R\J(R) such that two distinct vertices x and y are adjacent if and only if 1 - xy is not a unit of R. We obtain some graph theoretical properties of 𝔍R including its connectivity, planarity and perfectness and we compute some of its numerical invariants, namely diameter, girth, dominating number, independence number and vertex chromatic number and give an estimate for its edge chromatic number.

scholarly journals Clique Colourings of Geometric Graphs

10.37236/7159 ◽  
2018 ◽  
Vol 25 (4) ◽  
Author(s):  
Colin McDiarmid ◽  
Dieter Mitsche ◽  
Pawel Prałat
Keyword(s):  
Asymptotic Behaviour ◽  
Chromatic Number ◽  
Euclidean Distance ◽  
High Probability ◽  
Maximal Clique ◽  
Higher Dimensions ◽  
Geometric Graph ◽  
Geometric Graphs ◽  
Random Geometric Graph ◽  
Vertex Set

A clique colouring of a graph is a colouring of the vertices such that no maximal clique is monochromatic (ignoring isolated vertices). The least number of colours in such a colouring is the clique chromatic number.  Given $n$ points $\mathbf{x}_1, \ldots,\mathbf{x}_n$ in the plane, and a threshold $r>0$, the corresponding geometric graph has vertex set $\{v_1,\ldots,v_n\}$, and distinct $v_i$ and $v_j$ are adjacent when the Euclidean distance between $\mathbf{x}_i$ and $\mathbf{x}_j$ is at most $r$. We investigate the clique chromatic number of such graphs.We first show that the clique chromatic number is at most 9 for any geometric graph in the plane, and briefly consider geometric graphs in higher dimensions. Then we study the asymptotic behaviour of the clique chromatic number for the random geometric graph $\mathcal{G}$ in the plane, where $n$ random points are independently and uniformly distributed in a suitable square. We see that as $r$ increases from 0, with high probability the clique chromatic number is 1 for very small $r$, then 2 for small $r$, then at least 3 for larger $r$, and finally drops back to 2.

scholarly journals The Packing Chromatic Number of Different Jump Sizes of Circulant Graphs

2021 ◽  
Vol 33 (5) ◽  
pp. 66-73
Author(s):  
B. CHALUVARAJU ◽  
◽  
M. KUMARA ◽  
Keyword(s):  
Chromatic Number ◽  
Pairwise Distance ◽  
Circulant Graphs ◽  
Vertex Set ◽  
Packing Chromatic Number ◽  
Jump Sizes

The packing chromatic number χ_{p}(G) of a graph G = (V,E) is the smallest integer k such that the vertex set V(G) can be partitioned into disjoint classes V1 ,V2 ,...,Vk , where vertices in Vi have pairwise distance greater than i. In this paper, we compute the packing chromatic number of circulant graphs with different jump sizes._{}

The Classification of the Annihilating-Ideal Graphs of Commutative Rings

2014 ◽  
Vol 21 (02) ◽  
pp. 249-256 ◽  
Author(s):  
G. Aalipour ◽  
S. Akbari ◽  
M. Behboodi ◽  
R. Nikandish ◽  
M. J. Nikmehr ◽  
...  
Keyword(s):  
Commutative Ring ◽  
Chromatic Number ◽  
Artinian Ring ◽  
Clique Number ◽  
Commutative Rings ◽  
Vertex Set

Let R be a commutative ring and 𝔸(R) be the set of ideals with non-zero annihilators. The annihilating-ideal graph of R is defined as the graph 𝔸𝔾(R) with the vertex set 𝔸(R)* = 𝔸(R)\{(0)} and two distinct vertices I and J are adjacent if and only if IJ = (0). Here, we present some results on the clique number and the chromatic number of the annihilating-ideal graph of a commutative ring. It is shown that if R is an Artinian ring and ω (𝔸𝔾(R)) = 2, then R is Gorenstein. Also, we investigate commutative rings whose annihilating-ideal graphs are complete or bipartite.

A generalized ideal-based zero-divisor graph

2015 ◽  
Vol 14 (06) ◽  
pp. 1550079 ◽  
Author(s):  
M. J. Nikmehr ◽  
S. Khojasteh
Keyword(s):  
Prime Ideal ◽  
Commutative Ring ◽  
Chromatic Number ◽  
Zero Divisor ◽  
Undirected Graph ◽  
Clique Number ◽  
Proper Ideal ◽  
Graph Structure ◽  
Zero Divisor Graph ◽  
Vertex Set

Let R be a commutative ring with identity, I its proper ideal and M be a unitary R-module. In this paper, we introduce and study a kind of graph structure of an R-module M with respect to proper ideal I, denoted by ΓI(RM) or simply ΓI(M). It is the (undirected) graph with the vertex set M\{0} and two distinct vertices x and y are adjacent if and only if [x : M][y : M] ⊆ I. Clearly, the zero-divisor graph of R is a subgraph of Γ0(R); this is an important result on the definition. We prove that if ann R(M) ⊆ I and H is the subgraph of ΓI(M) induced by the set of all non-isolated vertices, then diam (H) ≤ 3 and gr (ΓI(M)) ∈ {3, 4, ∞}. Also, we prove that if Spec (R) and ω(Γ Nil (R)(M)) are finite, then χ(Γ Nil (R)(M)) ≤ ∣ Spec (R)∣ + ω(Γ Nil (R)(M)). Moreover, for a secondary R-module M and prime ideal P, we determine the chromatic number and the clique number of ΓP(M), where ann R(M) ⊆ P. Among other results, it is proved that for a semisimple R-module M with ann R(M) ⊆ I, ΓI(M) is a forest if and only if ΓI(M) is a union of isolated vertices or a star.

scholarly journals Commuting decomposition of Kn1,n2,...,nk through realization of the product A(G)A(GPk )

2018 ◽  
Vol 6 (1) ◽  
pp. 343-356
Author(s):  
K. Arathi Bhat ◽  
G. Sudhakara
Keyword(s):  
Chromatic Number ◽  
Perfect Matching ◽  
Domination Number ◽  
Perfect Matchings ◽  
Factor Graph ◽  
Vertex Set ◽  
Graph Parameters

Abstract In this paper, we introduce the notion of perfect matching property for a k-partition of vertex set of given graph. We consider nontrivial graphs G and GPk , the k-complement of graph G with respect to a kpartition of V(G), to prove that A(G)A(GPk ) is realizable as a graph if and only if P satis_es perfect matching property. For A(G)A(GPk ) = A(Γ) for some graph Γ, we obtain graph parameters such as chromatic number, domination number etc., for those graphs and characterization of P is given for which GPk and Γ are isomorphic. Given a 1-factor graph G with 2n vertices, we propose a partition P for which GPk is a graph of rank r and A(G)A(GPk ) is graphical, where n ≤ r ≤ 2n. Motivated by the result of characterizing decomposable Kn,n into commuting perfect matchings [2], we characterize complete k-partite graph Kn1,n2,...,nk which has a commuting decomposition into a perfect matching and its k-complement.

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