scholarly journals PEWARNAAN TITIK TOTAL SUPER ANTI-AJAIB LOKAL PADA GRAF PETERSEN DIPERUMUM P(n,k) DENGAN k=1,2

2021 ◽  
Vol 15 (4) ◽  
pp. 651-658
Author(s):  
Deddy Setyawan ◽  
Anis Nur Afni ◽  
Rafiantika Megahnia Prihandini ◽  
Ermita Rizki Albirri ◽  
Arika Indah Kristiana
Keyword(s):  
Natural Number ◽  
Chromatic Number ◽  
Petersen Graph ◽  
Vertex Coloring ◽  
Generalized Petersen Graph ◽  
Minimum Number ◽  
Vertex Label

The local antimagic total vertex labeling of graph G is a labeling that every vertices and edges label by natural number from 1 to  such that every two adjacent vertices has different weights, where is The sum of a vertex label and the labels of all edges that incident to the vertex. If the labeling start the smallest label from the vertex  then the edge  so that kind of coloring is called the local super antimagic total vertex labeling. That local super antimagic total vertex labeling induces vertex coloring of graph G where for vertex v, the weight  w(v) is the color of  v. The minimum number of colors that obtained by coloring that induces by local super antimagic total vertex labeling of G called the chromatic number of local super antimagic total vertex coloring of G, denoted by χlsat(G). In this paper, we consider the chromatic number of local super antimagic total vertex coloring of Generalized Petersen Graph P(n,k) for k=1, 2.

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 Local antimagic vertex coloring of unicyclic graphs

2018 ◽  
Vol 2 (1) ◽  
pp. 30 ◽  
Author(s):  
Nuris Hisan Nazula ◽  
S Slamin ◽  
D Dafik
Keyword(s):  
Chromatic Number ◽  
Vertex Coloring ◽  
Unicyclic Graphs ◽  
Antimagic Labeling ◽  
Minimum Number ◽  
Proper Vertex

The local antimagic labeling on a graph G with |V| vertices and |E| edges is defined to be an assignment f : E --&gt; {1, 2,..., |E|} so that the weights of any two adjacent vertices u and v are distinct, that is, w(u)̸  ̸= w(v) where w(u) = Σe∈<sub>E(u)</sub> f(e) and E(u) is the set of edges incident to u. Therefore, any local antimagic labeling induces a proper vertex coloring of G where the vertex u is assigned the color w(u). The local antimagic chromatic number, denoted by χla(G), is the minimum number of colors taken over all colorings induced by local antimagic labelings of G. In this paper, we present the local antimagic chromatic number of unicyclic graphs that is the graphs containing exactly one cycle such as kite and cycle with two neighbour pendants.

scholarly journals The Local Antimagic Chromatic Numbers of Some Join Graphs

2021 ◽  
Vol 26 (4) ◽  
pp. 80
Author(s):  
Xue Yang ◽  
Hong Bian ◽  
Haizheng Yu ◽  
Dandan Liu
Keyword(s):  
Partial Solution ◽  
Vertex Coloring ◽  
Chromatic Numbers ◽  
Infinite Class ◽  
Antimagic Labeling ◽  
Join Graph ◽  
Complement Graph ◽  
Minimum Number ◽  
Vertex Label ◽  
Proper Vertex

Let G=(V(G),E(G)) be a connected graph with n vertices and m edges. A bijection f:E(G)→{1,2,⋯,m} is an edge labeling of G. For any vertex x of G, we define ω(x)=∑e∈E(x)f(e) as the vertex label or weight of x, where E(x) is the set of edges incident to x, and f is called a local antimagic labeling of G, if ω(u)≠ω(v) for any two adjacent vertices u,v∈V(G). It is clear that any local antimagic labelling of G induces a proper vertex coloring of G by assigning the vertex label ω(x) to any vertex x of G. The local antimagic chromatic number of G, denoted by χla(G), is the minimum number of different vertex labels taken over all colorings induced by local antimagic labelings of G. In this paper, we present explicit local antimagic chromatic numbers of Fn∨K2¯ and Fn−v, where Fn is the friendship graph with n triangles and v is any vertex of Fn. Moreover, we explicitly construct an infinite class of connected graphs G such that χla(G)=χla(G∨K2¯), where G∨K2¯ is the join graph of G and the complement graph of complete graph K2. This fact leads to a counterexample to a theorem of Arumugam et al. in 2017, and our result also provides a partial solution to Problem 3.19 in Lau et al. in 2021.

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}).

scholarly journals Split-critical and uniquely split-colorable graphs

2010 ◽  
Vol Vol. 12 no. 5 (Graph and Algorithms) ◽  
Author(s):  
Tınaz Ekim ◽  
Bernard Ries ◽  
Dominique De Werra
Keyword(s):  
Chromatic Number ◽  
Point Of View ◽  
Vertex Coloring ◽  
Research Directions ◽  
Coloring Problem ◽  
Minimum Number ◽  
Vertex Coloring Problem ◽  
International Audience ◽  
Split Graphs

Graphs and Algorithms International audience The split-coloring problem is a generalized vertex coloring problem where we partition the vertices into a minimum number of split graphs. In this paper, we study some notions which are extensively studied for the usual vertex coloring and the cocoloring problem from the point of view of split-coloring, such as criticality and the uniqueness of the minimum split-coloring. We discuss some properties of split-critical and uniquely split-colorable graphs. We describe constructions of such graphs with some additional properties. We also study the effect of the addition and the removal of some edge sets on the value of the split-chromatic number. All these results are compared with their cochromatic counterparts. We conclude with several research directions on the topic.

scholarly journals Local Antimagic Chromatic Number for Copies of Graphs

Mathematics ◽  
2021 ◽  
Vol 9 (11) ◽  
pp. 1230
Author(s):  
Martin Bača ◽  
Andrea Semaničová-Feňovčíková ◽  
Tao-Ming Wang
Keyword(s):  
Chromatic Number ◽  
Disjoint Union ◽  
Vertex Coloring ◽  
Vertex Weight ◽  
Antimagic Labeling ◽  
Multiple Copies ◽  
Minimum Number ◽  
Proper Vertex ◽  
Edge Labeling

An edge labeling of a graph G=(V,E) using every label from the set {1,2,⋯,|E(G)|} exactly once is a local antimagic labeling if the vertex-weights are distinct for every pair of neighboring vertices, where a vertex-weight is the sum of labels of all edges incident with that vertex. Any local antimagic labeling induces a proper vertex coloring of G where the color of a vertex is its vertex-weight. This naturally leads to the concept of a local antimagic chromatic number. The local antimagic chromatic number is defined to be the minimum number of colors taken over all colorings of G induced by local antimagic labelings of G. In this paper, we estimate the bounds of the local antimagic chromatic number for disjoint union of multiple copies of a graph.

scholarly journals The local antimagic on disjoint union of some family graphs

2019 ◽  
Vol 5 (2) ◽  
pp. 69-75
Author(s):  
Marsidi Marsidi ◽  
Ika Hesti Agustin
Keyword(s):  
Lower Bound ◽  
Natural Number ◽  
Chromatic Number ◽  
Disjoint Union ◽  
Undirected Graph ◽  
Chromatic Numbers ◽  
Minimum Number ◽  
Vertex Set ◽  
Proper Vertex ◽  
Vertex Colouring

A graph  in this paper is nontrivial, finite, connected, simple, and undirected. Graph  consists of a vertex set and edge set. Let u,v be two elements in vertex set, and q is the cardinality of edge set in G, a bijective function from the edge set to the first q natural number is called a vertex local antimagic edge labelling if for any two adjacent vertices and , the weight of  is not equal with the weight of , where the weight of  (denoted by ) is the sum of labels of edges that are incident to . Furthermore, any vertex local antimagic edge labelling induces a proper vertex colouring on where  is the colour on the vertex . The vertex local antimagic chromatic number  is the minimum number of colours taken over all colourings induced by vertex local antimagic edge labelling of . In this paper, we discuss about the vertex local antimagic chromatic number on disjoint union of some family graphs, namely path, cycle, star, and friendship, and also determine the lower bound of vertex local antimagic chromatic number of disjoint union graphs. The chromatic numbers of disjoint union graph in this paper attend the lower bound.

Star chromatic number of Harary graphs

2020 ◽  
pp. 2150069
Author(s):  
Chitra Suseendran ◽  
Fathima Tabrez
Keyword(s):  
Chromatic Number ◽  
Vertex Coloring ◽  
Minimum Number ◽  
Star Coloring ◽  
Proper Vertex

A proper vertex coloring of a graph [Formula: see text] is called a star coloring if every path on four vertices in [Formula: see text] is not 2-colored. The star chromatic number is the minimum number of colors required to star color [Formula: see text] and it is denoted by [Formula: see text]. In this paper, the star coloring of Harary graphs [Formula: see text], where [Formula: see text] is even and [Formula: see text] is odd, is discussed.

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