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Abstract

Among difference vertex labelings of graphs, $\alpha$-labelings are the most restrictive one. A graph is an $\alpha$-graph if it admits an $\alpha$-labeling. In this work, we study a new alternative to construct $\alpha$-graphs using, the well-known, series-parallel operations on smaller $\alpha$-graphs. As an application of the series operation, we show that all members of a subfamily of all trees with maximum degree 4, obtained using vertex amalgamation of copies of the path $P_{11}$, are $\alpha$-graphs. We also show that the one-point union of up to four copies of $K_{n,n}$ is an $\alpha$-graph. In addition we prove that any $\alpha$-graph of order $m$ and size $n$ is an induced subgraph of a graph of order $m+2$ and size $m+n$. Furthermore, we prove that the Cartesian product of the bipartite graph $K_{2,n}$ and the path $P_m$ is an $\alpha$-graph.

Creative Commons License

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

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