R. Ichishima, S.C. Lˇpez , F.A. Muntaner-Batle, A. Oshima
On the beta-number of forests with isomorphic components
Discussiones Mathematicae Graph Theory
Received 27.07.2016, Revised 30.01.2017, Accepted 30.01.2017, doi: 10.7151/dmgt.2033

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 \left| f( u) -f( v) \right| and the resulting set of edge labels is { c,c+1,... ,c+\left| E( G) \right| -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 β s( 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\left| V( G) \right| -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.
beta-number, strong beta-number, graceful labeling, Skolem sequence, hooked Skolem sequence