Geodesic
On the Beta-Number of Forests with Isomorphic Components
Discussiones Mathematicae. Graph Theory, Tome 38 (2018) no. 3, pp. 683-701

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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.
Keywords: beta-number, strong beta-number, graceful labeling, Skolem sequence, hooked Skolem sequence 
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Ichishima, Rikio; López, Susana-Clara; Muntaner-Batle, Francesc Antoni; Oshima, Akito. On the Beta-Number of Forests with Isomorphic Components. Discussiones Mathematicae. Graph Theory, Tome 38 (2018) no. 3, pp. 683-701. http://geodesic.mathdoc.fr/item/DMGT_2018_38_3_a5/