scholarly journals PEWARNAAN PADA GRAF BINTANG SIERPINSKI

2017 ◽  
Vol 9 (1) ◽  
pp. 37
Author(s):  
Siti Khabibah
Keyword(s):  
Chromatic Number ◽  
Edge Coloring ◽  
Vertex Coloring ◽  
Star Graph ◽  
Sierpinski Triangle ◽  
Vertex Set

This paper discuss about Sierpinski star graph , which its construction based on the Sierpinski triangle. Vertex set of Sierpinski star graph  is a set of all triangles in Sierpinski triangle; and the edge set of Sierpinski star graph is a set of  all  sides that are joint edges of  two triangles on Sierpinski triangle. From the vertex and edge coloring of Sierpinski star graph, it is found that the chromatic number on vertex coloring of graph  is 1 for n = 1 and 2 for ; while the chromatic number on edge coloring of graf    is 0 for n = 1 and  for

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 Planar Mixed Hypergraphs

10.37236/1579 ◽  
2001 ◽  
Vol 8 (1) ◽  
Author(s):  
Zdeněk Dvořák ◽  
Daniel Král'
Keyword(s):  
Chromatic Number ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Vertex Coloring ◽  
Incidence Graph ◽  
Feasible Set ◽  
Vertex Set ◽  
Bridgeless Graph ◽  
Np Complete ◽  
Mixed Hypergraph

A mixed hypergraph $H$ is a triple $(V,{\cal C},{\cal D})$ where $V$ is its vertex set and ${\cal C}$ and ${\cal D}$ are families of subsets of $V$, ${\cal C}$–edges and ${\cal D}$–edges. A mixed hypergraph is a bihypergraph iff ${\cal C}={\cal D}$. A hypergraph is planar if its bipartite incidence graph is planar. A vertex coloring of $H$ is proper if each ${\cal C}$–edge contains two vertices with the same color and each ${\cal D}$–edge contains two vertices with different colors. The set of all $k$'s for which there exists a proper coloring using exactly $k$ colors is the feasible set of $H$; the feasible set is called gap-free if it is an interval. The minimum (maximum) number of the feasible set is called a lower (upper) chromatic number. We prove that the feasible set of any planar mixed hypergraph without edges of size two and with an edge of size at least four is gap-free. We further prove that a planar mixed hypergraph with at most two ${\cal D}$–edges of size two is two-colorable. We describe a polynomial-time algorithm to decide whether the lower chromatic number of a planar mixed hypergraph equals two. We prove that it is NP-complete to find the upper chromatic number of a mixed hypergraph even for 3-uniform planar bihypergraphs. In order to prove the latter statement, we prove that it is NP-complete to determine whether a planar 3-regular bridgeless graph contains a $2$-factor with at least a given number of components.

On the local irregularity vertex coloring of volcano, broom, parachute, double broom and complete multipartite graphs

2021 ◽  
Author(s):  
Arika Indah Kristiana ◽  
Nafidatun Nikmah ◽  
Dafik ◽  
Ridho Alfarisi ◽  
M. Ali Hasan ◽  
...  
Keyword(s):  
Chromatic Number ◽  
Connected Graph ◽  
Vertex Coloring ◽  
Minimum Cardinality ◽  
Label Function ◽  
Multipartite Graphs ◽  
Complete Multipartite Graphs ◽  
Vertex Set ◽  
Complete Multipartite ◽  
Local Irregularity

Let [Formula: see text] be a simple, finite, undirected, and connected graph with vertex set [Formula: see text] and edge set [Formula: see text]. A bijection [Formula: see text] is label function [Formula: see text] if [Formula: see text] and for any two adjacent vertices [Formula: see text] and [Formula: see text], [Formula: see text] where [Formula: see text] and [Formula: see text] is set ofvertices adjacent to [Formula: see text]. [Formula: see text] is called local irregularity vertex coloring. The minimum cardinality of local irregularity vertex coloring of [Formula: see text] is called chromatic number local irregular denoted by [Formula: see text]. In this paper, we verify the exact values of volcano, broom, parachute, double broom and complete multipartite graphs.

scholarly journals Local Super Antimagic Total Labeling for Vertex Coloring of Graphs

Symmetry ◽  
2020 ◽  
Vol 12 (11) ◽  
pp. 1843
Author(s):  
Slamin Slamin ◽  
Nelly Oktavia Adiwijaya ◽  
Muhammad Ali Hasan ◽  
Dafik Dafik ◽  
Kristiana Wijaya
Keyword(s):  
Chromatic Number ◽  
Vertex Coloring ◽  
Regular Graphs ◽  
Total Chromatic Number ◽  
Vertex Set ◽  
Tree Path ◽  
Injective Map ◽  
Proper Vertex

Let G=(V,E) be a graph with vertex set V and edge set E. A local antimagic total vertex coloring f of a graph G with vertex-set V and edge-set E is an injective map from V∪E to {1,2,…,|V|+|E|} such that if for each uv∈E(G) then w(u)≠w(v), where w(u)=∑uv∈E(G)f(uv)+f(u). If the range set f satisfies f(V)={1,2,…,|V|}, then the labeling is said to be local super antimagic total labeling. This labeling generates a proper vertex coloring of the graph G with the color w(v) assigning the vertex v. The local super antimagic total chromatic number of graph G, χlsat(G) is defined as the least number of colors that are used for all colorings generated by the local super antimagic total labeling of G. In this paper we investigate the existence of the local super antimagic total chromatic number for some particular classes of graphs such as a tree, path, cycle, helm, wheel, gear, sun, and regular graphs as well as an amalgamation of stars and an amalgamation of wheels.

scholarly journals Rainbow $H$-Factors of Complete $s$-Uniform $r$-Partite Hypergraphs

10.37236/901 ◽  
2008 ◽  
Vol 15 (1) ◽  
Author(s):  
Ailian Chen ◽  
Fuji Zhang ◽  
Hao Li
Keyword(s):  
Chromatic Number ◽  
Edge Coloring ◽  
Uniform Hypergraph ◽  
Vertex Partition ◽  
Proper Edge Coloring ◽  
Vertex Set ◽  
Positive Integers

We say a $s$-uniform $r$-partite hypergraph is complete, if it has a vertex partition $\{V_1,V_2,...,V_r\}$ of $r$ classes and its hyperedge set consists of all the $s$-subsets of its vertex set which have at most one vertex in each vertex class. We denote the complete $s$-uniform $r$-partite hypergraph with $k$ vertices in each vertex class by ${\cal T}_{s,r}(k)$. In this paper we prove that if $h,\ r$ and $s$ are positive integers with $2\leq s\leq r\leq h$ then there exists a constant $k=k(h,r,s)$ so that if $H$ is an $s$-uniform hypergraph with $h$ vertices and chromatic number $\chi(H)=r$ then any proper edge coloring of ${\cal T}_{s,r}(k)$ has a rainbow $H$-factor.

scholarly journals On The Local Edge Antimagic Coloring of Corona Product of Path and Cycle

CAUCHY ◽  
2019 ◽  
Vol 6 (1) ◽  
pp. 40
Author(s):  
Siti Aisyah ◽  
Ridho Alfarisi ◽  
Rafiantika M. Prihandini ◽  
Arika Indah Kristiana ◽  
Ratna Dwi Christyanti
Keyword(s):  
Chromatic Number ◽  
Connected Graph ◽  
Edge Coloring ◽  
Antimagic Labeling ◽  
Proper Edge Coloring ◽  
Minimum Number ◽  
Vertex Set ◽  
Local Edge ◽  
Corona Product

<p>Let  be a nontrivial and connected graph of vertex set  and edge set  . A bijection  is called a local edge antimagic labeling if for any two adjacent edges  and , where for . Thus, the local edge antimagic labeling induces a proper edge coloring of G if each edge e assigned the color  . The color of each an edge <em>e</em> = <em>uv</em> is assigned bywhich is defined by the sum of label both and vertices  and  . The local edge antimagic chromatic number, denoted by  is the minimum number of colors taken over all colorings induced by local edge antimagic labeling of   . In our paper, we present the local edge antimagic coloring of corona product of path and cycle, namely path corona cycle, cycle corona path, path corona path, cycle corona cycle.</p><p><strong>Keywords:</strong> Local antimagic; edge coloring; corona product; path; cycle.</p>

On the double total dominator chromatic number of graphs

2021 ◽  
pp. 2150154
Author(s):  
Fairouz Beggas ◽  
Hamamache Kheddouci ◽  
Walid Marweni
Keyword(s):  
Chromatic Number ◽  
Minimum Degree ◽  
Domination Number ◽  
Vertex Coloring ◽  
Total Domination Number ◽  
Close Relationship ◽  
Minimum Number ◽  
Number Formula ◽  
Dominator Coloring ◽  
Proper Vertex

In this paper, we introduce and study a new coloring problem of graphs called the double total dominator coloring. A double total dominator coloring of a graph [Formula: see text] with minimum degree at least 2 is a proper vertex coloring of [Formula: see text] such that each vertex has to dominate at least two color classes. The minimum number of colors among all double total dominator coloring of [Formula: see text] is called the double total dominator chromatic number, denoted by [Formula: see text]. Therefore, we establish the close relationship between the double total dominator chromatic number [Formula: see text] and the double total domination number [Formula: see text]. We prove the NP-completeness of the problem. We also examine the effects on [Formula: see text] when [Formula: see text] is modified by some operations. Finally, we discuss the [Formula: see text] number of square of trees by giving some bounds.

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.

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