Rikio Ichishima

The beta-number of a graph $G$ is the smallest positive integer $n$ for which there exists an injective function $f:V\left(G\right)\to \left\{0,1,\dots ,n\right\}$ such that each $uv\in E\left(G\right)$ is labeled $|f\left(u\right)-f\left(v\right)|$ and the resulting set of edge labels is $\left\{c,c+1,\dots ,\right.\left.c+\left|E\left(G\right)\right|-1\right\}$ for some positive integer $c$. The beta-number of $G$ is $+\infty$, 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.