On the strong beta-number of galaxies with three and four components
The beta-number of a graph is the smallest positive integer for which there exists an injective function such that each is labeled and the resulting set of edge labels is for some positive integer . The beta-number of is if there exists no such integer . If , then the resulting beta-number is called...
| Main Authors: | , , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
Taylor & Francis Group
2020-01-01
|
| Series: | AKCE International Journal of Graphs and Combinatorics |
| Subjects: | |
| Online Access: | http://dx.doi.org/10.1016/j.akcej.2019.03.004 |
| _version_ | 1819122072990777344 |
|---|---|
| author | Rikio Ichishima Francesc A. Muntaner-Batle Akito Oshima |
| author_facet | Rikio Ichishima Francesc A. Muntaner-Batle Akito Oshima |
| author_sort | Rikio Ichishima |
| collection | DOAJ |
| description | The beta-number of a graph is the smallest positive integer for which there exists an injective function such that each is labeled and the resulting set of edge labels is for some positive integer . The beta-number of is if there exists no such integer . If , then the resulting beta-number is called the strong beta-number of . A galaxy is a forest for which each component is a star. In this paper, we establish a lower bound for the strong beta-number of an arbitrary galaxy under certain conditions. We also determine formulas for the (strong) beta-number and gracefulness of galaxies with three and four components. As corollaries of these results, we provide formulas for the beta-number and gracefulness of the disjoint union of multiple copies of the same galaxies if the number of copies is odd. Based on this work, we propose some problems and a new conjecture. |
| first_indexed | 2024-12-22T06:46:39Z |
| format | Article |
| id | doaj.art-cd5faf321fae468f8c0ab9d2cb61e249 |
| institution | Directory Open Access Journal |
| issn | 0972-8600 2543-3474 |
| language | English |
| last_indexed | 2024-12-22T06:46:39Z |
| publishDate | 2020-01-01 |
| publisher | Taylor & Francis Group |
| record_format | Article |
| series | AKCE International Journal of Graphs and Combinatorics |
| spelling | doaj.art-cd5faf321fae468f8c0ab9d2cb61e2492022-12-21T18:35:16ZengTaylor & Francis GroupAKCE International Journal of Graphs and Combinatorics0972-86002543-34742020-01-0117131432310.1016/j.akcej.2019.03.0041759377On the strong beta-number of galaxies with three and four componentsRikio Ichishima0Francesc A. Muntaner-Batle1Akito Oshima2Department of Sport and Physical Education, Faculty of Physical Education, Kokushikan UniversityGraph Theory and Applications Research Group, School of Electrical Engineering and Computer Science, Faculty of Engineering and Built Environment, The University of NewcastleGraph Theory and Applications Research Group, School of Electrical Engineering and Computer Science, Faculty of Engineering and Built Environment, The University of NewcastleThe beta-number of a graph is the smallest positive integer for which there exists an injective function such that each is labeled and the resulting set of edge labels is for some positive integer . The beta-number of is if there exists no such integer . If , then the resulting beta-number is called the strong beta-number of . A galaxy is a forest for which each component is a star. In this paper, we establish a lower bound for the strong beta-number of an arbitrary galaxy under certain conditions. We also determine formulas for the (strong) beta-number and gracefulness of galaxies with three and four components. As corollaries of these results, we provide formulas for the beta-number and gracefulness of the disjoint union of multiple copies of the same galaxies if the number of copies is odd. Based on this work, we propose some problems and a new conjecture.http://dx.doi.org/10.1016/j.akcej.2019.03.004beta-numberstrong beta-numbergraph labeling-valuationgraceful labeling |
| spellingShingle | Rikio Ichishima Francesc A. Muntaner-Batle Akito Oshima On the strong beta-number of galaxies with three and four components AKCE International Journal of Graphs and Combinatorics beta-number strong beta-number graph labeling -valuation graceful labeling |
| title | On the strong beta-number of galaxies with three and four components |
| title_full | On the strong beta-number of galaxies with three and four components |
| title_fullStr | On the strong beta-number of galaxies with three and four components |
| title_full_unstemmed | On the strong beta-number of galaxies with three and four components |
| title_short | On the strong beta-number of galaxies with three and four components |
| title_sort | on the strong beta number of galaxies with three and four components |
| topic | beta-number strong beta-number graph labeling -valuation graceful labeling |
| url | http://dx.doi.org/10.1016/j.akcej.2019.03.004 |
| work_keys_str_mv | AT rikioichishima onthestrongbetanumberofgalaxieswiththreeandfourcomponents AT francescamuntanerbatle onthestrongbetanumberofgalaxieswiththreeandfourcomponents AT akitooshima onthestrongbetanumberofgalaxieswiththreeandfourcomponents |