On the beta-number of linear forests with an even number of 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 , otherwise. If , then the resulting beta-number is called the strong beta-number of . A linear forest is a forest for which each component is a path. In this paper, we determine a formula for the (strong) beta-number of the linear forests with two components. This leads us to a partial formula for the beta-number of the disjoint union of multiple copies of the same linear forests.

Keywords

• beta-number
• strong beta-number
• gracefulness
• graceful labeling
• linear forest