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

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 \(P_{t-1}\).