On the beta-number of linear forests with an even number of components

Abstract

The beta-number of a graph G is the smallest positive integer n for which there exists an injective function f:VG→0,1,…,n such that each uv∈EG is labeled |fu−fv| and the resulting set of edge labels is c,c+1,…,c+|EG|−1 for some positive integer c. The beta-number of G is +∞, otherwise. If c=1, then the resulting beta-number is called the strong beta-number of G. 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.