Dominator and Total Dominator Colorings in Graphs

Henning, Michael A.

doi:10.1007/978-3-030-58892-2_5

Cited by 6 publications

(5 citation statements)

References 14 publications

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“…A dominator coloring in which the open neighbourhood of every vertex contains at least one color class, is called a total dominator coloring (TD-coloring) and the minimum number of colors required for a TD-coloring is called the total dominator chromatic number of 𝐺, denoted by 𝜒 𝑑 𝑡 (𝐺) 5 . Further studies on 𝜒 𝑑 𝑡 (𝐺) and its variations can be found in 6,7 .…”

Section: Dominator Coloring Of Graphsmentioning

confidence: 99%

“…Some of the results which are significant and relevant in this study are listed below: Theorem 1 9 For any graph 𝐺 which admits tgdcoloring, 𝜒 𝑡𝑔𝑑 (𝐺) ≤ 𝛾 𝑡𝑔 (𝐺) + 𝜒(𝐺). Observation 1 6 If 𝑇 is a tree of order 𝑛 ≥ 2, then 𝛾 𝑡 (𝑇) ≤ 𝜒 𝑑 𝑡 (𝑇) ≤ 𝛾 𝑡 (𝑇) + 2.…”

Section: Dominator Coloring Of Graphsmentioning

confidence: 99%

“…𝑡 (𝑇) by 𝜒 𝑡𝑔𝑑 (𝑇) in the results corresponding in6 . For trees of diameter 3 and 4 can seen in Theorem 13.…”

mentioning

confidence: 99%

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Total Global Dominator Coloring of Trees and Unicyclic Graphs

Chithra

Joseph

2023

Baghdad Sci.J

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A total global dominator coloring of a graph is a proper vertex coloring of with respect to which every vertex in dominates a color class, not containing and does not dominate another color class. The minimum number of colors required in such a coloring of is called the total global dominator chromatic number, denoted by . In this paper, the total global dominator chromatic number of trees and unicyclic graphs are explored.

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Section: Dominator Coloring Of Graphsmentioning

confidence: 99%

Section: Dominator Coloring Of Graphsmentioning

confidence: 99%

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Total Global Dominator Coloring of Trees and Unicyclic Graphs

Chithra

Joseph

2023

Baghdad Sci.J

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“…e dominator chromatic number of G, as described by Henning and Michael [11], is the smallest number of colours among all dominator colourings of G. A suitable colouring of the vertices of a graph G is a total dominator colouring in which each vertex of the graph dominates every vertex of a colour class other than its own. G represents the total dominator chromatic number is the smallest number of colours in all of G is entire dominator colouring.…”

Section: Introductionmentioning

confidence: 99%

An Extensive Study on Graph Colourings and Dominator Chromatic Number of Sugeno-Type Fuzzy Graphs

Devi

2022

Mathematical Problems in Engineering

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The concept of graph colouring has become a very active field of research that enhances many practical applications and theoretical challenges. An accurate (vertex) colouring through the condition that each vertex is whichever unaccompanied in its colour class or adjoining to all vertices of at least one colour class is known as dominator colouring. A graph is dominator colouring is a suitable colouring in which at least one vertex dominates each colour class. Many difficult and global issues are solved using fuzzy graph colouring approaches. In this paper, we introduce a new Sugeno-Type Fuzzy graph of groups, which is found using several fuzzy graph operations such as dominant path-colouring number, cycle, union, join, and products. A number that is representative based on those vertices in the formations of all paths with those vertices as their starts and ends to compare with other paths is the minimum number of shared edges based on those vertices in the formations of all paths with those vertices as their starts and ends to compare with other paths. The sugeno dominant path-colouring number is the minimum Sugeno number of shared edges among the Sugeno cardinality of all sets of shared edges, which allows for various techniques. These results are used in studying various and recently introduced chromatic numbers of graphs.

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“…As examples we can cite dominator coloring [5], total dominator coloring [7], and dominated coloring [11]. A survey of selected results on dominator and total dominator colorings can be found in the book chapter [8]. Consider a k-coloring C of G with color classes V 1 , V 2 , .…”

mentioning

confidence: 99%

Global dominated coloring of graphs

Kalarkop¹,

Hamid²,

Chellali³

et al. 2024

Discuss. Math. Graph Theory

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In this paper, we initiate a study of global dominated coloring of graphs as a variation of dominated colorings. A global dominated coloring of a graph G is a proper coloring such that for each color class there are at least two vertices, one of which is adjacent to all the vertices of this class while the other one is not adjacent to any vertex of the class. The global dominated chromatic number of G is the minimum number of colors used among all global dominated colorings of G. In this paper, we establish various bounds on the global dominated chromatic number of a graph in terms of some graph invariants including the order, dominated chromatic number, domination number and total domination number. Moreover, characterizations of extremal graphs attaining some of these bounds are provided. We also discuss the global dominated coloring in trees and split graphs.

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Dominator and Total Dominator Colorings in Graphs

Cited by 6 publications

References 14 publications

Total Global Dominator Coloring of Trees and Unicyclic Graphs

Total Global Dominator Coloring of Trees and Unicyclic Graphs

An Extensive Study on Graph Colourings and Dominator Chromatic Number of Sugeno-Type Fuzzy Graphs

Global dominated coloring of graphs

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