On the strong beta-number of galaxies with three and four components

AKCE International Journal of Graphs and Combinatorics

Rikio Ichishima¹  Francesc A. Muntaner-Batle²  Akito Oshima² 

[1] Department of Sport and Physical Education, Faculty of Physical Education, Kokushikan University;Graph Theory and Applications Research Group, School of Electrical Engineering and Computer Science, Faculty of Engineering and Built Environment, The University of Newcastle;

关键词: beta-number;    strong beta-number;    graph labeling;    -valuation;    graceful labeling;   

DOI  :  10.1016/j.akcej.2019.03.004

来源: DOAJ

【 摘 要 】

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.

【 授权许可】

Unknown