The strong 3-rainbow index of some certain graphs and its amalgamation
We introduce a strong \(k\)-rainbow index of graphs as modification of well-known \(k\)-rainbow index of graphs. A tree in an edge-colored connected graph \(G\), where adjacent edge may be colored the same, is a rainbow tree if all of its edges have distinct colors. Let \(k\) be an integer with \(2\...
| Main Authors: | , |
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| Format: | Article |
| Language: | English |
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AGH Univeristy of Science and Technology Press
2022-06-01
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| Series: | Opuscula Mathematica |
| Subjects: | |
| Online Access: | https://www.opuscula.agh.edu.pl/vol42/4/art/opuscula_math_4223.pdf |
| _version_ | 1818242265544392704 |
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| author | Zata Yumni Awanis A.N.M. Salman |
| author_facet | Zata Yumni Awanis A.N.M. Salman |
| author_sort | Zata Yumni Awanis |
| collection | DOAJ |
| description | We introduce a strong \(k\)-rainbow index of graphs as modification of well-known \(k\)-rainbow index of graphs. A tree in an edge-colored connected graph \(G\), where adjacent edge may be colored the same, is a rainbow tree if all of its edges have distinct colors. Let \(k\) be an integer with \(2\leq k\leq n\). The strong \(k\)-rainbow index of \(G\), denoted by \(srx_k(G)\), is the minimum number of colors needed in an edge-coloring of \(G\) so that every \(k\) vertices of \(G\) is connected by a rainbow tree with minimum size. We focus on \(k=3\). We determine the strong \(3\)-rainbow index of some certain graphs. We also provide a sharp upper bound for the strong \(3\)-rainbow index of amalgamation of graphs. Additionally, we determine the exact values of the strong \(3\)-rainbow index of amalgamation of some graphs. |
| first_indexed | 2024-12-12T13:42:29Z |
| format | Article |
| id | doaj.art-8a2cf1583353467f86ebf5e9aed4da2c |
| institution | Directory Open Access Journal |
| issn | 1232-9274 |
| language | English |
| last_indexed | 2024-12-12T13:42:29Z |
| publishDate | 2022-06-01 |
| publisher | AGH Univeristy of Science and Technology Press |
| record_format | Article |
| series | Opuscula Mathematica |
| spelling | doaj.art-8a2cf1583353467f86ebf5e9aed4da2c2022-12-22T00:22:45ZengAGH Univeristy of Science and Technology PressOpuscula Mathematica1232-92742022-06-01424527547https://doi.org/10.7494/OpMath.2022.42.4.5274223The strong 3-rainbow index of some certain graphs and its amalgamationZata Yumni Awanis0https://orcid.org/0000-0001-8927-6043A.N.M. Salman1Institut Teknologi Bandung, Combinatorial Mathematics Research Group, Faculty of Mathematics and Natural Sciences, Jalan Ganesa 10, Bandung 40132, IndonesiaInstitut Teknologi Bandung, Combinatorial Mathematics Research Group, Faculty of Mathematics and Natural Sciences, Jalan Ganesa 10, Bandung 40132, IndonesiaWe introduce a strong \(k\)-rainbow index of graphs as modification of well-known \(k\)-rainbow index of graphs. A tree in an edge-colored connected graph \(G\), where adjacent edge may be colored the same, is a rainbow tree if all of its edges have distinct colors. Let \(k\) be an integer with \(2\leq k\leq n\). The strong \(k\)-rainbow index of \(G\), denoted by \(srx_k(G)\), is the minimum number of colors needed in an edge-coloring of \(G\) so that every \(k\) vertices of \(G\) is connected by a rainbow tree with minimum size. We focus on \(k=3\). We determine the strong \(3\)-rainbow index of some certain graphs. We also provide a sharp upper bound for the strong \(3\)-rainbow index of amalgamation of graphs. Additionally, we determine the exact values of the strong \(3\)-rainbow index of amalgamation of some graphs.https://www.opuscula.agh.edu.pl/vol42/4/art/opuscula_math_4223.pdfamalgamationrainbow coloringrainbow steiner treestrong \(k\)-rainbow index |
| spellingShingle | Zata Yumni Awanis A.N.M. Salman The strong 3-rainbow index of some certain graphs and its amalgamation Opuscula Mathematica amalgamation rainbow coloring rainbow steiner tree strong \(k\)-rainbow index |
| title | The strong 3-rainbow index of some certain graphs and its amalgamation |
| title_full | The strong 3-rainbow index of some certain graphs and its amalgamation |
| title_fullStr | The strong 3-rainbow index of some certain graphs and its amalgamation |
| title_full_unstemmed | The strong 3-rainbow index of some certain graphs and its amalgamation |
| title_short | The strong 3-rainbow index of some certain graphs and its amalgamation |
| title_sort | strong 3 rainbow index of some certain graphs and its amalgamation |
| topic | amalgamation rainbow coloring rainbow steiner tree strong \(k\)-rainbow index |
| url | https://www.opuscula.agh.edu.pl/vol42/4/art/opuscula_math_4223.pdf |
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