ON LOCAL IRREGULARITY OF THE VERTEX COLORING OF THE CORONA PRODUCT OF A TREE GRAPH
Let \(G=(V,E)\) be a graph with a vertex set \(V\) and an edge set \(E\). The graph \(G\) is said to be with a local irregular vertex coloring if there is a function \(f\) called a local irregularity vertex coloring with the properties: (i) \(l:(V(G)) \to \{ 1,2,...,k \} \) as a vertex irregular \(k...
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Ural Branch of the Russian Academy of Sciences and Ural Federal University named after the first President of Russia B.N.Yeltsin, Krasovskii Institute of Mathematics and Mechanics
2022-12-01
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| Series: | Ural Mathematical Journal |
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| Online Access: | https://umjuran.ru/index.php/umj/article/view/391 |
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| author | Arika Indah Kristiana M. Hidayat Robiatul Adawiyah D. Dafik Susi Setiawani Ridho Alfarisi |
| author_facet | Arika Indah Kristiana M. Hidayat Robiatul Adawiyah D. Dafik Susi Setiawani Ridho Alfarisi |
| author_sort | Arika Indah Kristiana |
| collection | DOAJ |
| description | Let \(G=(V,E)\) be a graph with a vertex set \(V\) and an edge set \(E\). The graph \(G\) is said to be with a local irregular vertex coloring if there is a function \(f\) called a local irregularity vertex coloring with the properties: (i) \(l:(V(G)) \to \{ 1,2,...,k \} \) as a vertex irregular \(k\)-labeling and \(w:V(G)\to N,\) for every \(uv \in E(G),\) \({w(u)\neq w(v)}\) where \(w(u)=\sum_{v\in N(u)}l(i)\) and (ii) \(\mathrm{opt}(l)=\min\{ \max \{ l_{i}: l_{i} \ \text{is a vertex irregular labeling}\}\}\). The chromatic number of the local irregularity vertex coloring of \(G\) denoted by \(\chi_{lis}(G)\), is the minimum cardinality of the largest label over all such local irregularity vertex colorings. In this paper, we study a local irregular vertex coloring of \(P_m\bigodot G\) when \(G\) is a family of tree graphs, centipede \(C_n\), double star graph \((S_{2,n})\), Weed graph \((S_{3,n})\), and \(E\) graph \((E_{3,n})\). |
| format | Article |
| id | doaj-art-bb674d9a115c4b97a9838c9a330634f8 |
| institution | Kabale University |
| issn | 2414-3952 |
| language | English |
| publishDate | 2022-12-01 |
| publisher | Ural Branch of the Russian Academy of Sciences and Ural Federal University named after the first President of Russia B.N.Yeltsin, Krasovskii Institute of Mathematics and Mechanics |
| record_format | Article |
| series | Ural Mathematical Journal |
| spelling | doaj-art-bb674d9a115c4b97a9838c9a330634f82025-08-20T03:39:26ZengUral Branch of the Russian Academy of Sciences and Ural Federal University named after the first President of Russia B.N.Yeltsin, Krasovskii Institute of Mathematics and MechanicsUral Mathematical Journal2414-39522022-12-018210.15826/umj.2022.2.008161ON LOCAL IRREGULARITY OF THE VERTEX COLORING OF THE CORONA PRODUCT OF A TREE GRAPHArika Indah Kristiana0M. Hidayat1Robiatul Adawiyah2D. Dafik3Susi Setiawani4Ridho Alfarisi5Department of Mathematics Education, University of Jember, Jalan Kalimantan 37, 68126, Jember, Jawa TimurDepartment of Mathematics Education, University of Jember, Jalan Kalimantan 37, 68126, Jember, Jawa TimurDepartment of Mathematics Education, University of Jember, Jalan Kalimantan 37, 68126, Jember, Jawa TimurDepartment of Mathematics Education, University of Jember, Jalan Kalimantan 37, 68126, Jember, Jawa TimurDepartment of Mathematics Education, University of Jember, Jalan Kalimantan 37, 68126, Jember, Jawa TimurDepartment of Elementary School Education, University of Jember, Jalan Kalimantan 37, 68126, Jember, Jawa TimurLet \(G=(V,E)\) be a graph with a vertex set \(V\) and an edge set \(E\). The graph \(G\) is said to be with a local irregular vertex coloring if there is a function \(f\) called a local irregularity vertex coloring with the properties: (i) \(l:(V(G)) \to \{ 1,2,...,k \} \) as a vertex irregular \(k\)-labeling and \(w:V(G)\to N,\) for every \(uv \in E(G),\) \({w(u)\neq w(v)}\) where \(w(u)=\sum_{v\in N(u)}l(i)\) and (ii) \(\mathrm{opt}(l)=\min\{ \max \{ l_{i}: l_{i} \ \text{is a vertex irregular labeling}\}\}\). The chromatic number of the local irregularity vertex coloring of \(G\) denoted by \(\chi_{lis}(G)\), is the minimum cardinality of the largest label over all such local irregularity vertex colorings. In this paper, we study a local irregular vertex coloring of \(P_m\bigodot G\) when \(G\) is a family of tree graphs, centipede \(C_n\), double star graph \((S_{2,n})\), Weed graph \((S_{3,n})\), and \(E\) graph \((E_{3,n})\).https://umjuran.ru/index.php/umj/article/view/391local irregularity, corona product, tree graph family. |
| spellingShingle | Arika Indah Kristiana M. Hidayat Robiatul Adawiyah D. Dafik Susi Setiawani Ridho Alfarisi ON LOCAL IRREGULARITY OF THE VERTEX COLORING OF THE CORONA PRODUCT OF A TREE GRAPH Ural Mathematical Journal local irregularity, corona product, tree graph family. |
| title | ON LOCAL IRREGULARITY OF THE VERTEX COLORING OF THE CORONA PRODUCT OF A TREE GRAPH |
| title_full | ON LOCAL IRREGULARITY OF THE VERTEX COLORING OF THE CORONA PRODUCT OF A TREE GRAPH |
| title_fullStr | ON LOCAL IRREGULARITY OF THE VERTEX COLORING OF THE CORONA PRODUCT OF A TREE GRAPH |
| title_full_unstemmed | ON LOCAL IRREGULARITY OF THE VERTEX COLORING OF THE CORONA PRODUCT OF A TREE GRAPH |
| title_short | ON LOCAL IRREGULARITY OF THE VERTEX COLORING OF THE CORONA PRODUCT OF A TREE GRAPH |
| title_sort | on local irregularity of the vertex coloring of the corona product of a tree graph |
| topic | local irregularity, corona product, tree graph family. |
| url | https://umjuran.ru/index.php/umj/article/view/391 |
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