On the beta-number of forests with isomorphic components

Abstract

The beta-number, β (G), of a graph G is defined to be either the smallest positive integer n for which there exists an injective function f : V (G) → {0, 1, . . . , n} such that each uv ∈ E (G) is labeled |f (u) − f (v)| and the resulting set of edge labels is {c, c+ 1, . . . , c+|E (G)| −1} for some positive integer c or +∞ if there exists no such integer n. If c = 1, then the resulting beta-number is called the strong beta-number of G and is denoted by βs (G). In this paper, we show that if G is a bipartite graph and m is odd, then β (mG) ≤ mβ (G) + m − 1. This leads us to conclude that β (mG) = m |V (G)| − 1 if G has the additional property that G is a graceful nontrivial tree. In addition to these, we examine the (strong) beta-number of forests whose components are isomorphic to either paths or stars.