GRACEFUL CHROMATIC NUMBER OF SOME CARTESIAN PRODUCT GRAPHS
A graph \(G(V,E)\) is a system consisting of a finite non empty set of vertices \(V(G)\) and a set of edges \(E(G)\). A (proper) vertex colouring of \(G\) is a function \(f:V(G)\rightarrow \{1,2,\ldots,k\},\) for some positive integer \(k\) such that \(f(u)\neq f(v)\) for every edge \(uv\in E(G)\)....
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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
2023-12-01
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| Series: | Ural Mathematical Journal |
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| Online Access: | https://umjuran.ru/index.php/umj/article/view/600 |
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| author | I Nengah Suparta Mathiyazhagan Venkatachalam I Gede Aris Gunadi Putu Andi Cipta Pratama |
| author_facet | I Nengah Suparta Mathiyazhagan Venkatachalam I Gede Aris Gunadi Putu Andi Cipta Pratama |
| author_sort | I Nengah Suparta |
| collection | DOAJ |
| description | A graph \(G(V,E)\) is a system consisting of a finite non empty set of vertices \(V(G)\) and a set of edges \(E(G)\). A (proper) vertex colouring of \(G\) is a function \(f:V(G)\rightarrow \{1,2,\ldots,k\},\) for some positive integer \(k\) such that \(f(u)\neq f(v)\) for every edge \(uv\in E(G)\). Moreover, if \(|f(u)-f(v)|\neq |f(v)-f(w)|\) for every adjacent edges \(uv,vw\in E(G)\), then the function \(f\) is called graceful colouring for \(G\). The minimum number \(k\) such that \(f\) is a graceful colouring for \(G\) is called the graceful chromatic number of \(G\). The purpose of this research is to determine graceful chromatic number of Cartesian product graphs \(C_m \times P_n\) for integers \(m\geq 3\) and \(n\geq 2\), and \(C_m \times C_n\) for integers \(m,n\geq 3\). Here, \(C_m\) and \(P_m\) are cycle and path with \(m\) vertices, respectively. We found some exact values and bounds for graceful chromatic number of these mentioned Cartesian product graphs. |
| format | Article |
| id | doaj-art-71b1aec0130b44d3ba7d9f2fe004b06d |
| institution | DOAJ |
| issn | 2414-3952 |
| language | English |
| publishDate | 2023-12-01 |
| 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-71b1aec0130b44d3ba7d9f2fe004b06d2025-08-20T02:52:34ZengUral 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 Journal2414-39522023-12-019210.15826/umj.2023.2.016189GRACEFUL CHROMATIC NUMBER OF SOME CARTESIAN PRODUCT GRAPHSI Nengah Suparta0Mathiyazhagan Venkatachalam1I Gede Aris Gunadi2Putu Andi Cipta Pratama3Department of Mathematics, Universitas Pendidikan Ganesha, Jl. Udayana 11, Singaraja-Bali 81117Department of Mathematics, Kongunadu Arts and Science College, Coimbatore–641029, Tamil NaduDepartment of Mathematics, Universitas Pendidikan Ganesha, Jl. Udayana 11, Singaraja-Bali 81117Department of Mathematics, Universitas Pendidikan Ganesha, Jl. Udayana 11, Singaraja-Bali 81117A graph \(G(V,E)\) is a system consisting of a finite non empty set of vertices \(V(G)\) and a set of edges \(E(G)\). A (proper) vertex colouring of \(G\) is a function \(f:V(G)\rightarrow \{1,2,\ldots,k\},\) for some positive integer \(k\) such that \(f(u)\neq f(v)\) for every edge \(uv\in E(G)\). Moreover, if \(|f(u)-f(v)|\neq |f(v)-f(w)|\) for every adjacent edges \(uv,vw\in E(G)\), then the function \(f\) is called graceful colouring for \(G\). The minimum number \(k\) such that \(f\) is a graceful colouring for \(G\) is called the graceful chromatic number of \(G\). The purpose of this research is to determine graceful chromatic number of Cartesian product graphs \(C_m \times P_n\) for integers \(m\geq 3\) and \(n\geq 2\), and \(C_m \times C_n\) for integers \(m,n\geq 3\). Here, \(C_m\) and \(P_m\) are cycle and path with \(m\) vertices, respectively. We found some exact values and bounds for graceful chromatic number of these mentioned Cartesian product graphs.https://umjuran.ru/index.php/umj/article/view/600graceful colouring, graceful chromatics number, cartesian product graph |
| spellingShingle | I Nengah Suparta Mathiyazhagan Venkatachalam I Gede Aris Gunadi Putu Andi Cipta Pratama GRACEFUL CHROMATIC NUMBER OF SOME CARTESIAN PRODUCT GRAPHS Ural Mathematical Journal graceful colouring, graceful chromatics number, cartesian product graph |
| title | GRACEFUL CHROMATIC NUMBER OF SOME CARTESIAN PRODUCT GRAPHS |
| title_full | GRACEFUL CHROMATIC NUMBER OF SOME CARTESIAN PRODUCT GRAPHS |
| title_fullStr | GRACEFUL CHROMATIC NUMBER OF SOME CARTESIAN PRODUCT GRAPHS |
| title_full_unstemmed | GRACEFUL CHROMATIC NUMBER OF SOME CARTESIAN PRODUCT GRAPHS |
| title_short | GRACEFUL CHROMATIC NUMBER OF SOME CARTESIAN PRODUCT GRAPHS |
| title_sort | graceful chromatic number of some cartesian product graphs |
| topic | graceful colouring, graceful chromatics number, cartesian product graph |
| url | https://umjuran.ru/index.php/umj/article/view/600 |
| work_keys_str_mv | AT inengahsuparta gracefulchromaticnumberofsomecartesianproductgraphs AT mathiyazhaganvenkatachalam gracefulchromaticnumberofsomecartesianproductgraphs AT igedearisgunadi gracefulchromaticnumberofsomecartesianproductgraphs AT putuandiciptapratama gracefulchromaticnumberofsomecartesianproductgraphs |