A \emph{graceful labeling} of a graph $G$ of size $n$ is an injective assignment of integers from the set $\{0,1,\dots,n\}$ to the vertices of $G$ such that when each edge has assigned a \emph{weight}, given by the absolute value of the difference of the labels of its end vertices, all the weights are distinct. A graceful labeling is called an $\alpha$-labeling when the graph $G$ is bipartite, with stable sets $A$ and $B$, and the labels assigned to the vertices in $A$ are smaller than the labels assigned to the vertices in $B$. In this work we study graceful and $\alpha$-labelings of graphs. We prove that the Cartesian product of two $\alpha$-trees results in an $\alpha$-tree when both trees admit $\alpha$-labelings and their stable sets are balanced. In addition, we present a tree that has the property that when any number of pendant vertices are attached to the vertices of any subset of its smaller stable set, the resulting graph is an $\alpha$-tree. We also prove the existence of an $\alpha$-labeling of three types of graphs obtained by connecting, sequentially, any number of paths of equal size.

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This work is licensed under a Creative Commons Attribution 4.0 License.