New $\alpha$-Trees and Graceful Unions of $\alpha$-Graphs and Linear Forests

Abstract

In this paper, we study five methods to construct $\alpha$-trees by using vertex amalgamations of smaller $\alpha$-trees. We also study graceful and $\alpha$-labelings for graphs that are the union of $t$ copies of an $\alpha$-graph $G$ of order $m$ and size $n$ with a graph $H$ of size $t$. If $n>m$, we prove that the disjoint union of $H$ and $t$ copies of $G$ is graceful when $H$ is graceful, and that this union is an $\alpha$-graph when $H$ is any linear forest of size $t–1$. If $n=m$, we prove that this union is an $\alpha$-graph when $H$ is the path $P_{t-1}$.