scholarly journals BILANGAN KROMATIK LOKASI UNTUK JOIN DARI DUA GRAF

2013 ◽  
Vol 2 (1) ◽  
pp. 23
Author(s):  
Yuli Erita
Keyword(s):  
Chromatic Number ◽  
Connected Graph ◽  
Minimum Number

Let f be a proper k-coloring of a connected graph G and = (V) bean ordered partition of V (G) into the resulting color classes. For a vertex v of G, thecolor code of v with respect to is dened to be the ordered k-tuplec(v) = (d(v; V1); d(v; V2); :::; d(v; V));where d(v; Vi) = minfd(v; x)jx 2 Vikg, 1 i k: If distinct vertices have distinct colorcodes, then f is called a locating coloring. The minimum number of colors needed in alocating coloring of G is the locating chromatic number of G, and denoted by (G). Inthis paper, we study the locating chromatic number of the join of some graphs.

scholarly journals On local antimagic vertex coloring of corona products related to friendship and fan graph

2021 ◽  
Vol 5 (2) ◽  
pp. 110
Author(s):  
Zein Rasyid Himami ◽  
Denny Riama Silaban
Keyword(s):  
Chromatic Number ◽  
Connected Graph ◽  
Vertex Coloring ◽  
Antimagic Labeling ◽  
Minimum Number ◽  
Fan Graph ◽  
Corona Product

Let <em>G</em>=(<em>V</em>,<em>E</em>) be connected graph. A bijection <em>f </em>: <em>E</em> → {1,2,3,..., |<em>E</em>|} is a local antimagic of <em>G</em> if any adjacent vertices <em>u,v</em> ∈ <em>V</em> satisfies <em>w</em>(<em>u</em>)≠ <em>w</em>(<em>v</em>), where <em>w</em>(<em>u</em>)=∑<sub>e∈E(u) </sub><em>f</em>(<em>e</em>), <em>E</em>(<em>u</em>) is the set of edges incident to <em>u</em>. When vertex <em>u</em> is assigned the color <em>w</em>(<em>u</em>), we called it a local antimagic vertex coloring of <em>G</em>. A local antimagic chromatic number of <em>G</em>, denoted by <em>χ</em><sub>la</sub>(<em>G</em>), is the minimum number of colors taken over all colorings induced by the local antimagic labeling of <em>G</em>. In this paper, we determine the local antimagic chromatic number of corona product of friendship and fan with null graph on <em>m</em> vertices, namely, <em>χ</em><sub>la</sub>(<em>F</em><sub>n</sub> ⊙ \overline{K_m}) and <em>χ</em><sub>la</sub>(<em>f</em><sub>(1,n)</sub> ⊙ \overline{K_m}).

On the sigma chromatic number of the join of a finite number of paths and cycles

2019 ◽  
pp. 2150019
Author(s):  
Agnes D. Garciano ◽  
Maria Czarina T. Lagura ◽  
Reginaldo M. Marcelo
Keyword(s):  
Finite Number ◽  
Chromatic Number ◽  
Connected Graph ◽  
Odd Cycle ◽  
Minimum Number ◽  
Simple Connected Graph

Let [Formula: see text] be a simple connected graph and [Formula: see text] a coloring of the vertices in [Formula: see text] For any [Formula: see text], let [Formula: see text] be the sum of colors of the vertices adjacent to [Formula: see text]. Then [Formula: see text] is called a sigma coloring of [Formula: see text] if for any two adjacent vertices [Formula: see text] The minimum number of colors needed in a sigma coloring of [Formula: see text] is the sigma chromatic number of [Formula: see text], denoted by [Formula: see text] In this paper, we prescribe a sigma coloring of the join of paths and cycles. As a consequence, we determine the sigma chromatic number of the join of a finite number of paths and cycles. In particular, let [Formula: see text], where [Formula: see text] or [Formula: see text] with [Formula: see text] If [Formula: see text], where [Formula: see text] and [Formula: see text], then [Formula: see text] if [Formula: see text] is an odd cycle, for some [Formula: see text] and [Formula: see text] otherwise.

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>

Joins of paths and cycles of equal order with sigma chromatic number 2

2021 ◽  
pp. 2250006
Author(s):  
Agnes D. Garciano ◽  
Maria Czarina T. Lagura ◽  
Reginaldo M. Marcelo
Keyword(s):  
Chromatic Number ◽  
Connected Graph ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Odd Cycle ◽  
Necessary And Sufficient ◽  
Positive Integers ◽  
Simple Connected Graph

For a simple connected graph [Formula: see text] let [Formula: see text] be a coloring of [Formula: see text] where two adjacent vertices may be assigned the same color. Let [Formula: see text] be the sum of colors of neighbors of any vertex [Formula: see text] The coloring [Formula: see text] is a sigma coloring of [Formula: see text] if for any two adjacent vertices [Formula: see text] [Formula: see text] The least number of colors required in a sigma coloring of [Formula: see text] is the sigma chromatic number of [Formula: see text] and is denoted by [Formula: see text] A sigma coloring of a graph is a neighbor-distinguishing type of coloring and it is known that the sigma chromatic number of a graph is bounded above by its chromatic number. It is also known that for a path [Formula: see text] and a cycle [Formula: see text] where [Formula: see text] [Formula: see text] and [Formula: see text] if [Formula: see text] is even. Let [Formula: see text] the join of the graphs [Formula: see text], where [Formula: see text] or [Formula: see text] [Formula: see text] and [Formula: see text] is not an odd cycle for any [Formula: see text]. It has been shown that if [Formula: see text] for [Formula: see text] and [Formula: see text] then [Formula: see text]. In this study, we give necessary and sufficient conditions under which [Formula: see text] where [Formula: see text] is the join of copies of [Formula: see text] and/or [Formula: see text] for the same value of [Formula: see text]. Let [Formula: see text] and [Formula: see text] be positive integers with [Formula: see text] and [Formula: see text] In this paper, we show that [Formula: see text] if and only if [Formula: see text] or [Formula: see text] is odd, [Formula: see text] is even and [Formula: see text]; and [Formula: see text] if and only if [Formula: see text] is even and [Formula: see text] We also obtain necessary and sufficient conditions on [Formula: see text] and [Formula: see text], so that [Formula: see text] for [Formula: see text] where [Formula: see text] or [Formula: see text] other than the cases [Formula: see text] and [Formula: see text]

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.

scholarly journals Total Coloring Conjecture for Certain Classes of Graphs

Algorithms ◽  
2018 ◽  
Vol 11 (10) ◽  
pp. 161 ◽  
Author(s):  
R. Vignesh ◽  
J. Geetha ◽  
K. Somasundaram
Keyword(s):  
Chromatic Number ◽  
Maximum Degree ◽  
Line Graph ◽  
Product Line ◽  
Total Coloring ◽  
Lexicographic Product ◽  
Total Chromatic Number ◽  
Minimum Number ◽  
Total Coloring Conjecture

A total coloring of a graph G is an assignment of colors to the elements of the graph G such that no two adjacent or incident elements receive the same color. The total chromatic number of a graph G, denoted by χ ′ ′ ( G ) , is the minimum number of colors that suffice in a total coloring. Behzad and Vizing conjectured that for any graph G, Δ ( G ) + 1 ≤ χ ′ ′ ( G ) ≤ Δ ( G ) + 2 , where Δ ( G ) is the maximum degree of G. In this paper, we prove the total coloring conjecture for certain classes of graphs of deleted lexicographic product, line graph and double graph.

On the metric dimension and diameter of circulant graphs with three jumps

2018 ◽  
Vol 10 (01) ◽  
pp. 1850008
Author(s):  
Muhammad Imran ◽  
A. Q. Baig ◽  
Saima Rashid ◽  
Andrea Semaničová-Feňovčíková
Keyword(s):  
Positive Integer ◽  
Connected Graph ◽  
Metric Spaces ◽  
Metric Dimension ◽  
Resolving Set ◽  
Circulant Graphs ◽  
Infinite Class ◽  
Metric Properties ◽  
Minimum Number ◽  
Metric Basis

Let [Formula: see text] be a connected graph and [Formula: see text] be the distance between the vertices [Formula: see text] and [Formula: see text] in [Formula: see text]. The diameter of [Formula: see text] is defined as [Formula: see text] and is denoted by [Formula: see text]. A subset of vertices [Formula: see text] is called a resolving set for [Formula: see text] if for every two distinct vertices [Formula: see text], there is a vertex [Formula: see text], [Formula: see text], such that [Formula: see text]. A resolving set containing the minimum number of vertices is called a metric basis for [Formula: see text] and the number of vertices in a metric basis is its metric dimension, denoted by [Formula: see text]. Metric dimension is a generalization of affine dimension to arbitrary metric spaces (provided a resolving set exists). Let [Formula: see text] be a family of connected graphs [Formula: see text] depending on [Formula: see text] as follows: the order [Formula: see text] and [Formula: see text]. If there exists a constant [Formula: see text] such that [Formula: see text] for every [Formula: see text] then we shall say that [Formula: see text] has bounded metric dimension, otherwise [Formula: see text] has unbounded metric dimension. If all graphs in [Formula: see text] have the same metric dimension, then [Formula: see text] is called a family of graphs with constant metric dimension. In this paper, we study the metric properties of an infinite class of circulant graphs with three generators denoted by [Formula: see text] for any positive integer [Formula: see text] and when [Formula: see text]. We compute the diameter and determine the exact value of the metric dimension of these circulant graphs.

On strict strong coloring of graphs

2020 ◽  
pp. 2150040
Author(s):  
A. Mohammed Abid ◽  
T. R. Ramesh Rao
Keyword(s):  
Chromatic Number ◽  
Proper Coloring ◽  
Color Class ◽  
Minimum Number ◽  
Strong Chromatic Number

A strict strong coloring of a graph [Formula: see text] is a proper coloring of [Formula: see text] in which every vertex of the graph is adjacent to every vertex of some color class. The minimum number of colors required for a strict strong coloring of [Formula: see text] is called the strict strong chromatic number of [Formula: see text] and is denoted by [Formula: see text]. In this paper, we characterize the results on strict strong coloring of Mycielskian graphs and iterated Mycielskian graphs.

2-Distance chromatic number of some graph products

2020 ◽  
Vol 12 (02) ◽  
pp. 2050021
Author(s):  
Ghazale Ghazi ◽  
Freydoon Rahbarnia ◽  
Mostafa Tavakoli
Keyword(s):  
Chromatic Number ◽  
Lower Bounds ◽  
Upper And Lower Bounds ◽  
Graph Products ◽  
Lexicographic Product ◽  
Graph Product ◽  
Minimum Number

This paper studies the 2-distance chromatic number of some graph product. A coloring of [Formula: see text] is 2-distance if any two vertices at distance at most two from each other get different colors. The minimum number of colors in the 2-distance coloring of [Formula: see text] is the 2-distance chromatic number and denoted by [Formula: see text]. In this paper, we obtain some upper and lower bounds for the 2-distance chromatic number of the rooted product, generalized rooted product, hierarchical product and we determine exact value for the 2-distance chromatic number of the lexicographic product.

Analysis on the Component Connectivity of Enhanced Hypercubes

2020 ◽  
Author(s):  
Liqiong Xu ◽  
Litao Guo
Keyword(s):  
Connected Graph ◽  
Interconnection Networks ◽  
Unsolved Problem ◽  
Reliability Evaluation ◽  
Minimum Number ◽  
Appl Math

Abstract Reliability evaluation of interconnection networks is of significant importance to the design and maintenance of interconnection networks. The component connectivity is an important parameter for the reliability evaluation of interconnection networks and is a generalization of the traditional connectivity. The $g$-component connectivity $c\kappa _g (G)$ of a non-complete connected graph $G$ is the minimum number of vertices whose deletion results in a graph with at least $g$ components. Determining the $g$-component connectivity is still an unsolved problem in many interconnection networks. Let $Q_{n,k}$ ($1\leq k\leq n-1$) denote the $(n, k)$-enhanced hypercube. In this paper, let $n\geq 7$ and $1\leq k \leq n-5$, we determine $c\kappa _{g}(Q_{n,k}) = g(n + 1) - \frac{1}{2}g(g + 1) + 1$ for $2 \leq g \leq n$. The previous result in Zhao and Yang (2019, Conditional connectivity of folded hypercubes. Discret. Appl. Math., 257, 388–392) is extended.

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