The even vertex magic total labelings of t-fold wheels
Let $ G $ be a graph of order $ n $ and size $ m $. A vertex magic total labeling of $ G $ is a one-to-one function $ f $: $ V(G) \cup E(G) \rightarrow \{1, 2, \cdots, n+m\} $ with the property that for each vertex $ u $ of $ G $, the sum of the label of $ u $ and the labels of all edges incident to...
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| Format: | Članak |
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AIMS Press
2023-09-01
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| Serija: | AIMS Mathematics |
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| Online pristup: | https://www.aimspress.com/article/doi/10.3934/math.20231407?viewType=HTML |
| _version_ | 1827786268935716864 |
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| author | Supaporn Saduakdee Varanoot Khemmani |
| author_facet | Supaporn Saduakdee Varanoot Khemmani |
| author_sort | Supaporn Saduakdee |
| collection | DOAJ |
| description | Let $ G $ be a graph of order $ n $ and size $ m $. A vertex magic total labeling of $ G $ is a one-to-one function $ f $: $ V(G) \cup E(G) \rightarrow \{1, 2, \cdots, n+m\} $ with the property that for each vertex $ u $ of $ G $, the sum of the label of $ u $ and the labels of all edges incident to $ u $ is the same constant, referred to as the magic constant. Such a labeling is even if $ f[V(G)] = \{2, 4, 6, \cdots, 2n\} $. A graph $ G $ is called an even vertex magic if there is an even vertex magic total labeling of $ G $. The primary goal of this paper is to study wheel related graphs with the size greater than the order, which have an even vertex magic total labeling. For every integer $ n \geq 3 $ and $ t \geq 1 $, the $ t $-fold wheel $ W_{n, t} $ is a wheel related graph derived from a wheel $ W_n $ by duplicating the $ t $ hubs, each adjacent to all rim vertices, and not adjacent to each other. The $ t $-fold wheel $ W_{n, t} $ has a size $ nt + n $ that exceeds its order $ n + t $. In this paper, we determine the magic constant of the $ t $-fold wheel $ W_{n, t} $, the bound of an integer $ t $ for the even vertex magic total labeling of the $ t $-fold wheel $ W_{n, t} $ and the conditions for even vertex magic $ W_{n, t} $, focusing on integers $ n $ and $ t $ are established. Additionally, we investigate the necessary conditions for the even vertex magic total labeling of the $ n $-fold wheel $ W_{n, n} $ when $ n $ is odd and the $ n $-fold wheel $ W_{n, n-2} $ when $ n $ is even. Furthermore, our study explores the characterization of an even vertex magic $ W_{n, t} $ for integer $ 3 \leq n \leq 9 $. |
| first_indexed | 2024-03-11T16:32:06Z |
| format | Article |
| id | doaj.art-4b9c9d3350d146f6b404d13118284ed8 |
| institution | Directory Open Access Journal |
| issn | 2473-6988 |
| language | English |
| last_indexed | 2024-03-11T16:32:06Z |
| publishDate | 2023-09-01 |
| publisher | AIMS Press |
| record_format | Article |
| series | AIMS Mathematics |
| spelling | doaj.art-4b9c9d3350d146f6b404d13118284ed82023-10-24T01:23:09ZengAIMS PressAIMS Mathematics2473-69882023-09-01811275132752710.3934/math.20231407The even vertex magic total labelings of t-fold wheelsSupaporn Saduakdee0Varanoot Khemmani11. Program of Mathematics, Chandrakasem Rajabhat University, Ratchadaphisek Road, Bangkok 10900, Thailand2. Department of Mathematics, Srinakharinwirot University, Sukhumvit 23, Bangkok 10110, ThailandLet $ G $ be a graph of order $ n $ and size $ m $. A vertex magic total labeling of $ G $ is a one-to-one function $ f $: $ V(G) \cup E(G) \rightarrow \{1, 2, \cdots, n+m\} $ with the property that for each vertex $ u $ of $ G $, the sum of the label of $ u $ and the labels of all edges incident to $ u $ is the same constant, referred to as the magic constant. Such a labeling is even if $ f[V(G)] = \{2, 4, 6, \cdots, 2n\} $. A graph $ G $ is called an even vertex magic if there is an even vertex magic total labeling of $ G $. The primary goal of this paper is to study wheel related graphs with the size greater than the order, which have an even vertex magic total labeling. For every integer $ n \geq 3 $ and $ t \geq 1 $, the $ t $-fold wheel $ W_{n, t} $ is a wheel related graph derived from a wheel $ W_n $ by duplicating the $ t $ hubs, each adjacent to all rim vertices, and not adjacent to each other. The $ t $-fold wheel $ W_{n, t} $ has a size $ nt + n $ that exceeds its order $ n + t $. In this paper, we determine the magic constant of the $ t $-fold wheel $ W_{n, t} $, the bound of an integer $ t $ for the even vertex magic total labeling of the $ t $-fold wheel $ W_{n, t} $ and the conditions for even vertex magic $ W_{n, t} $, focusing on integers $ n $ and $ t $ are established. Additionally, we investigate the necessary conditions for the even vertex magic total labeling of the $ n $-fold wheel $ W_{n, n} $ when $ n $ is odd and the $ n $-fold wheel $ W_{n, n-2} $ when $ n $ is even. Furthermore, our study explores the characterization of an even vertex magic $ W_{n, t} $ for integer $ 3 \leq n \leq 9 $.https://www.aimspress.com/article/doi/10.3934/math.20231407?viewType=HTMLeven vertex magic total labelingeven vertex magicwheelt-fold wheelwheel related graph |
| spellingShingle | Supaporn Saduakdee Varanoot Khemmani The even vertex magic total labelings of t-fold wheels AIMS Mathematics even vertex magic total labeling even vertex magic wheel t-fold wheel wheel related graph |
| title | The even vertex magic total labelings of t-fold wheels |
| title_full | The even vertex magic total labelings of t-fold wheels |
| title_fullStr | The even vertex magic total labelings of t-fold wheels |
| title_full_unstemmed | The even vertex magic total labelings of t-fold wheels |
| title_short | The even vertex magic total labelings of t-fold wheels |
| title_sort | even vertex magic total labelings of t fold wheels |
| topic | even vertex magic total labeling even vertex magic wheel t-fold wheel wheel related graph |
| url | https://www.aimspress.com/article/doi/10.3934/math.20231407?viewType=HTML |
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