On the beta-number of forests with isomorphic components

Abstract

The beta-number, ß (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 +8 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 ßs (G). In this paper, we show that if G is a bipartite graph and m is odd, then ß (mG) = mß (G) + m - 1. This leads us to conclude that ß (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.