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

Abstract

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.

Keywords

• beta-number
• strong beta-number
• graph labeling
• -valuation
• graceful labeling