Bilangan Kromatik Pewarnaan Titik pada Graf Dual dari Graf Roda

Penelitian ini bertujuan mengkonstruksi graf dual dari graf roda (Wn*) dan menentukan bilangan kromatik graf dual dari graf roda (Wn*). Penelitian ini dimulai dari menggambarkan beberapa graf roda dari ke , kemudian membangun graf dual dari graf roda dengan memanfaatkan graf-graf dari ke , kemudian memberikan warna pada titik-titik dari graf dualnya dengan menentukan bilangan kromatiknya. Diperoleh hasil bahwa Graf roda merupakan graf self-dual karena isomorfik dengan graf dualnya yaitu . Pewarnaan titik diperoleh dengan menentukan bilangan kromatik graf dual dari graf roda, menentukan pola dari bilangan kromatik, dan memberikan warna. Berdasarkan hasil penelitian, diperoleh bilangan kromatik pewarnaan titik pada graf dual dari graf roda yakni Kata Kunci: Pewarnaan Titik, Bilangan Kromatik, Graf Dual dan Graf Roda.This research aims to construct a dual graph from a wheel graph (Wn*) and determine the dual graph chromatic number of the wheel graph (Wn*). This research starts from describing some wheel graph from to , then construct a dual graph from a wheel graph from to , then gives color to the vertices of the dual graph by determining the chromatic number. The result showed that the wheel graph is a self-dual graph because it is isomorphic with its dual graph, namely . The vertex coloring is obtained by determining the chromatic number of the dual graph of the wheel graph, determining the pattern of the chromatic number and giving the color. Based on the research results, the chromatic number of vertex coloring on dual graph of a wheel graph is: Keywords: Vertex Coloring, Chromatic Number, Dual Graph and Wheel Graph.

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On the double total dominator chromatic number of graphs

Discrete Mathematics Algorithms and Applications ◽

10.1142/s1793830921501548 ◽

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.

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Distinguishability of Locally Finite Trees

The Electronic Journal of Combinatorics ◽

10.37236/947 ◽

2007 ◽

Vol 14 (1) ◽

Cited By ~ 11

Author(s):

Mark E. Watkins ◽

Xiangqian Zhou

Keyword(s):

Positive Integer ◽

Chromatic Number ◽

Inverse Image ◽

Vertex Coloring ◽

Finite Tree ◽

Locally Finite ◽

Distinguishing Number ◽

Nonidentity Element ◽

Locally Countable ◽

Proper Vertex

The distinguishing number $\Delta(X)$ of a graph $X$ is the least positive integer $n$ for which there exists a function $f:V(X)\to\{0,1,2,\cdots,n-1\}$ such that no nonidentity element of $\hbox{Aut}(X)$ fixes (setwise) every inverse image $f^{-1}(k)$, $k\in\{0,1,2,\cdots,n-1\}$. All infinite, locally finite trees without pendant vertices are shown to be 2-distinguishable. A proof is indicated that extends 2-distinguishability to locally countable trees without pendant vertices. It is shown that every infinite, locally finite tree $T$ with finite distinguishing number contains a finite subtree $J$ such that $\Delta(J)=\Delta(T)$. Analogous results are obtained for the distinguishing chromatic number, namely the least positive integer $n$ such that the function $f$ is also a proper vertex-coloring.

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On the b-chromatic number of some graph products

Studia Scientiarum Mathematicarum Hungarica ◽

10.1556/sscmath.49.2012.2.1194 ◽

2012 ◽

Vol 49 (2) ◽

pp. 156-169 ◽

Cited By ~ 4

Author(s):

Marko Jakovac ◽

Iztok Peterin

Keyword(s):

Chromatic Number ◽

Upper And Lower Bounds ◽

Vertex Coloring ◽

Graph Products ◽

Lexicographic Product ◽

Arbitrary Graph ◽

Strong Product ◽

Color Class ◽

Complete Bipartite Graphs ◽

Proper Vertex

A b-coloring is a proper vertex coloring of a graph such that each color class contains a vertex that has a neighbor in all other color classes and the b-chromatic number is the largest integer φ(G) for which a graph has a b-coloring with φ(G) colors. We determine some upper and lower bounds for the b-chromatic number of the strong product G ⊠ H, the lexicographic product G[H] and the direct product G × H and give some exact values for products of paths, cycles, stars, and complete bipartite graphs. We also show that the b-chromatic number of Pn ⊠ H, Cn ⊠ H, Pn[H], Cn[H], and Km,n[H] can be determined for an arbitrary graph H, when integers m and n are large enough.

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A Study on Harmonious Coloring of Circulant Networks

International Journal of Engineering & Technology ◽

10.14419/ijet.v7i4.10.20945 ◽

2018 ◽

Vol 7 (4.10) ◽

pp. 393

Author(s):

Franklin Thamil Selvi.M.S ◽

Amutha A ◽

Antony Mary A

Keyword(s):

Chromatic Number ◽

Simple Graph ◽

Vertex Coloring ◽

Circulant Networks ◽

Proper Vertex

Given a simple graph , a harmonious coloring of is the proper vertex coloring such that each pair of colors seems to appears together on at most one edge. The harmonious chromatic number of , denoted by is the minimal number of colors in a harmonious coloring of . In this paper we have determined the harmonious chromatic number of some classes of Circulant Networks.

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Local antimagic vertex coloring of unicyclic graphs

Indonesian Journal of Combinatorics ◽

10.19184/ijc.2018.2.1.4 ◽

2018 ◽

Vol 2 (1) ◽

pp. 30 ◽

Cited By ~ 3

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 --> {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∈E(u) 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.

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Exact 2-Distance b-Coloring and Exact 2-Distance b-Continuity of Helm Graph 𝑯𝒏

YMER Digital ◽

10.37896/ymer20.10/8 ◽

2021 ◽

Vol 20 (10) ◽

pp. 62-72

Author(s):

S Saraswathi ◽

◽

M Poobalaranjani ◽

Keyword(s):

Chromatic Number ◽

Vertex Coloring ◽

Class A ◽

Color Class

An exact 2-distance coloring of a graph 𝐺 is a coloring of vertices of 𝐺 such that any two vertices which are at distance exactly 2 receive distinct colors. An exact 2-distance chromatic number𝑒2(𝐺) of 𝐺 is the minimum 𝑘 for which 𝐺 admits an exact 2-distance coloring with 𝑘 colors. A 𝑏-coloring of 𝐺 by 𝑘 colors is a proper 𝑘-vertex coloring such that in each color class, there exists a vertex called a color dominating vertex which has a neighbor in every other color class. A vertex that has a 2-neighbor in all other color classes is called an exact 2-distance color dominating vertex (or an 𝑒2-cdv). Exact 2-distance 𝑏-coloring (or an 𝑒2𝑏-coloring) of 𝐺 is an exact 2-distance coloring such that each color class contains an 𝑒2- cdv. An exact 2-distance 𝑏-chromatic number (or an 𝑒2𝑏-number) 𝑒2𝑏(𝐺) of 𝐺 is the largest integer 𝑘 such that 𝐺 has an 𝑒2𝑏-coloring with 𝑘colors. If for each integer𝑘, 𝑒2(𝐺) ≤ 𝑘 ≤ 𝑒2𝑏(𝐺), 𝐺 has an 𝑒2𝑏-coloring by 𝑘 colors, then 𝐺 is said to be an exact 2-distance 𝑏- continuous graph. In this paper, the 𝑒2𝑏-number𝑒2𝑏(𝐻𝑛)of the helm graph 𝐻𝑛is obtained and 𝑒2𝑏-continuity of 𝐻𝑛is discussed.

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A Study on Radio Labeling of Diameter N-2 and Caterpillar Graphs

International Journal of Emerging Research in Management and Technology ◽

10.23956/ijermt.v6i8.115 ◽

2018 ◽

Vol 6 (8) ◽

pp. 30

Author(s):

M. Shakila ◽

N. Rajakumari

Keyword(s):

Chromatic Number ◽

Basic Type ◽

Graph Labeling ◽

Vertex Coloring ◽

Not Given ◽

Minimum Value ◽

Caterpillar Graphs ◽

Radio Labeling

Radio labeling of graphs is a specific type of graph labeling. The basic type of graph labeling is vertex coloring; this is where the vertices of a graph G are assigned different colors so that adjacent vertices are not given the same color. A k-coloring of a graph G is a coloring that uses k colors. The chromatic number of a graph G is the minimum value for k such that a k-coloring exists for G [2].

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Star chromatic number of some graphs

Discrete Mathematics Algorithms and Applications ◽

10.1142/s1793830921500890 ◽

2021 ◽

pp. 2150089

Author(s):

S. Akbari ◽

M. CHAVOOSHI ◽

M. Ghanbari ◽

S. Taghian

Keyword(s):

Chromatic Number ◽

Cartesian Product ◽

Vertex Coloring ◽

Star Coloring ◽

Proper Vertex

A proper vertex coloring of a graph [Formula: see text] is called a star coloring if every two color classes induce a forest whose each component is a star, which means there is no bicolored [Formula: see text] in [Formula: see text]. In this paper, we show that the Cartesian product of any two cycles, except [Formula: see text] and [Formula: see text], has a [Formula: see text]-star coloring.

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b-Chromatic sum of a graph

Discrete Mathematics Algorithms and Applications ◽

10.1142/s1793830915500408 ◽

2015 ◽

Vol 07 (04) ◽

pp. 1550040 ◽

Cited By ~ 2

Author(s):

P. C. Lisna ◽

M. S. Sunitha

Keyword(s):

Bipartite Graph ◽

Chromatic Number ◽

Complete Graph ◽

Complete Bipartite Graph ◽

Double Star ◽

Star Graph ◽

Proper Coloring ◽

Color Class ◽

Wheel Graph ◽

Chromatic Sum

A b-coloring of a graph G is a proper coloring of the vertices of G such that there exists a vertex in each color class joined to at least one vertex in each other color classes. The b-chromatic number of a graph G, denoted by [Formula: see text], is the maximum integer [Formula: see text] such that G admits a b-coloring with [Formula: see text] colors. In this paper we introduce a new concept, the b-chromatic sum of a graph [Formula: see text], denoted by [Formula: see text] and is defined as the minimum of sum of colors [Formula: see text] of [Formula: see text] for all [Formula: see text] in a b-coloring of [Formula: see text] using [Formula: see text] colors. Also obtained the b-chromatic sum of paths, cycles, wheel graph, complete graph, star graph, double star graph, complete bipartite graph, corona of paths and corona of cycles.

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

Indonesian Journal of Combinatorics ◽

10.19184/ijc.2021.5.2.7 ◽

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

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