On the Independent Coloring of Graphs with Applications to the Independence Number of Cartesian Product Graphs
Let G be a graph with V=VG. A nonempty subset S of V is called an independent set of G if no two distinct vertices in S are adjacent. The union of a class {S:S is an independent set of G} and ∅ is denoted by IG. For a graph H, a function f:V⟶IH is called an H−independent coloring of G (or simply cal...
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| Format: | Article |
| Language: | English |
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Wiley
2023-01-01
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| Series: | Journal of Mathematics |
| Online Access: | http://dx.doi.org/10.1155/2023/2601205 |
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| _version_ | 1849307383693574144 |
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| author | Nopparat Pleanmani Sayan Panma Nuttawoot Nupo |
| author_facet | Nopparat Pleanmani Sayan Panma Nuttawoot Nupo |
| author_sort | Nopparat Pleanmani |
| collection | DOAJ |
| description | Let G be a graph with V=VG. A nonempty subset S of V is called an independent set of G if no two distinct vertices in S are adjacent. The union of a class {S:S is an independent set of G} and ∅ is denoted by IG. For a graph H, a function f:V⟶IH is called an H−independent coloring of G (or simply called an H− coloring) if fx∩fy=∅ for any adjacent vertices x,y∈V and fV is a class of disjoint sets. Let αH,G denote the maximum cardinality of the set{∑x∈Vfx:f is an H− coloring of G}. In this paper, we obtain basic properties of an H− coloring of G and find αH,G of some families of graphs G and H. Furthermore, we apply them to determine the independence number of the Cartesian product of a complete graph Kn and a graph G and prove that αKn□G=αKn,G. |
| format | Article |
| id | doaj-art-2937a9905bfe4485aae4d19cd005a101 |
| institution | Kabale University |
| issn | 2314-4785 |
| language | English |
| publishDate | 2023-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Journal of Mathematics |
| spelling | doaj-art-2937a9905bfe4485aae4d19cd005a1012025-08-20T03:54:47ZengWileyJournal of Mathematics2314-47852023-01-01202310.1155/2023/2601205On the Independent Coloring of Graphs with Applications to the Independence Number of Cartesian Product GraphsNopparat Pleanmani0Sayan Panma1Nuttawoot Nupo2Department of MathematicsAdvanced Research Center for Computational SimulationDepartment of MathematicsLet G be a graph with V=VG. A nonempty subset S of V is called an independent set of G if no two distinct vertices in S are adjacent. The union of a class {S:S is an independent set of G} and ∅ is denoted by IG. For a graph H, a function f:V⟶IH is called an H−independent coloring of G (or simply called an H− coloring) if fx∩fy=∅ for any adjacent vertices x,y∈V and fV is a class of disjoint sets. Let αH,G denote the maximum cardinality of the set{∑x∈Vfx:f is an H− coloring of G}. In this paper, we obtain basic properties of an H− coloring of G and find αH,G of some families of graphs G and H. Furthermore, we apply them to determine the independence number of the Cartesian product of a complete graph Kn and a graph G and prove that αKn□G=αKn,G.http://dx.doi.org/10.1155/2023/2601205 |
| spellingShingle | Nopparat Pleanmani Sayan Panma Nuttawoot Nupo On the Independent Coloring of Graphs with Applications to the Independence Number of Cartesian Product Graphs Journal of Mathematics |
| title | On the Independent Coloring of Graphs with Applications to the Independence Number of Cartesian Product Graphs |
| title_full | On the Independent Coloring of Graphs with Applications to the Independence Number of Cartesian Product Graphs |
| title_fullStr | On the Independent Coloring of Graphs with Applications to the Independence Number of Cartesian Product Graphs |
| title_full_unstemmed | On the Independent Coloring of Graphs with Applications to the Independence Number of Cartesian Product Graphs |
| title_short | On the Independent Coloring of Graphs with Applications to the Independence Number of Cartesian Product Graphs |
| title_sort | on the independent coloring of graphs with applications to the independence number of cartesian product graphs |
| url | http://dx.doi.org/10.1155/2023/2601205 |
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