Some Results on T-Coloring and St-Coloring of Generalized Butterfly Graphs

Moran, Rubul; Bora, Niranjan; Pegu, Aditya; Chamua, Monjit

doi:10.17654/0974165822022

Cited by 2 publications

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“…Let the graph G = (V, E) be given and T be a finite set of non negative integers. The T −coloring and ST -coloring of G are defined in the works of [1,16]. Application of T and ST −coloring of graph naturally arises in the modeling of different scientific problems.…”

Section: Introductionmentioning

confidence: 99%

On <img src=image/13430241_01.gif>-coloring and <img src=image/13430241_02.gif>-coloring ofWindmill Graph

Moran¹,

Bora²,

Bhattacharyya³

2023

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The windmill graph W (r, m); m ≥ 3, r ≥ 2 is the graph formed by joining a common vertex to every vertex of m copies of the complete graph K r . T -coloring of a graph is a map h defined on the set of vertices in such a way that for any edge (w 1 , w 2 ), | h(w 1 ) − h(w 2 ) | does not belong to a finite set T of non-negative integers. Strong T-Coloring (ST -coloring) is a particular case of T -coloring and is defined as the map:for any two distinct edges (w 1 , w 2 ), (w 3 , w 4 ) ∈ E(G). Application of T and ST -coloring of graph naturally arises in the modeling of different scientific problems. Frequency assignment problem (FAP) is one of the well known problems in the field of telecommunication, which can be modeled using the concept of T and ST -coloring of graphs. In this paper, we will consider two special types of T -sets. The first one is λ-initial set, introduced by Cozzens and Roberts, which is of the form {0, 1, 2, . . . , λ} ∪ S where S is any arbitrary set that doesn't contain any multiple of (λ + 1). The second one is λ-multiple of q set, introduced by Raychaudhuri, which is of the form {0, q, 2q, . . . , λq} ∪ S, where S is a subset of the set {q + 1, q + 2, q + 3, . . . , λq}. We will discuss some parameters related to these two types of colorings viz. T -chromatic number, T -span, T -edge span on the basis of the two T -sets. We will also deduce some generalized results of ST -coloring of any graph based on any T -set, and with the help of these results we will obtain ST -chromatic number and bounds for the ST -span and ST -edge span of windmill graphs.

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Section: Introductionmentioning

confidence: 99%

On <img src=image/13430241_01.gif>-coloring and <img src=image/13430241_02.gif>-coloring ofWindmill Graph

Moran¹,

Bora²,

Bhattacharyya³

2023

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“…The study on domination and coloring in wrapped butterfly and bloom graphs are interesting areas of research in graph theory. In wrapped butterfly network, few studies like 1-harmonious coloring (1) , T-coloring (2) , ST-coloring (2) , total domination (3) and domination of its digraph (4) were carried on. In the above research work, only one concept, either coloring or domination were applied, whereas in this paper both the concepts are applied together to determine the dom-chromatic number of wrapped butterfly graph.…”

Section: Introductionmentioning

confidence: 99%

Dom-Chromatic Number o f Wrapped Butterfly & Bloom Graphs

Punitha,

Angeline

2023

IJST

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Objectives: For a given graph G with proper coloring, the problem of selecting a dom-coloring set is to choose a dominating set having a property that it has a minimum of one vertex from every possible color class in G. Our aim is to determine the family of networks that allow dom-coloring and to find its dom-chromatic number denoted by γ dc (G). Method: We have applied the algorithmic method of choosing the dom-coloring set(dc-set). Here we have designed a coloring algorithm to yield the proper coloring for the vertices of the graph. D-set algorithm has been developed to determine the dominating set for the given graph. Then, the dc-set for the graph is obtained by applying the above two algorithms. Findings: In this study, we have established the study on finding the dc-set of wrapped butterfly network and bloom graphs. Further, we have found the dom-chromatic number of the above-mentioned graphs. Novelty: Dom-coloring is an extended variation of graph coloring and domination which has emerged as a result of the combination of the two broad concepts in graph theory namely, domination and coloring. A dominating set which includes a minimum of one vertex from all possible color classes of the graph forms a dom-coloring set. In this paper, a study on dom-coloring of wrapped butterfly and bloom graphs have been accomplished. These results may be generalized for butterfly derived networks to determine its domchromatic number.

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Some Results on T-Coloring and St-Coloring of Generalized Butterfly Graphs

Cited by 2 publications

References 0 publications

On <img src=image/13430241_01.gif>-coloring and <img src=image/13430241_02.gif>-coloring ofWindmill Graph

On <img src=image/13430241_01.gif>-coloring and <img src=image/13430241_02.gif>-coloring ofWindmill Graph

Dom-Chromatic Number o f Wrapped Butterfly & Bloom Graphs

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