A new application of the ${\otimes }_h$-product to $\alpha$-labelings

The weak tensor product was introduced by Snevily as a way to construct new graphs that admit $\alpha$-labelings from a pair of known $\alpha$-graphs. In this article, we show that this product and the application to $\alpha$-labelings can be generalized by considering as a second factor of the product, a family $\mathrm{\Gamma }$ of bipartite $(p,q)$-graphs, $p$ and $q$ fixed. The only additional restriction that we should consider is that for every $F\in \Gamma$,thereexistsand$\alpha$-labeling$f_F$with$f_F(V(F))=L\cup H$, where $L,H\subset [0,q]$ are the stable sets induced by the characteristic of $f_F$ and they do not depend on $F$. We also obtain analogous applications to near $\alpha$-labelings and bigraceful labelings.