A new application of the ⊗h⊗h-product to αα-labelings

The weak tensor product was introduced by Snevily as a way to construct new graphs that admit αα-labelings from a pair of known αα-graphs. In this article, we show that this product and the application to αα-labelings can be generalized by considering as a second factor of the product, a family ΓΓ of bipartite (p,q)(p,q)-graphs, pp and qq fixed. The only additional restriction that we should consider is that for every F∈ΓF∈Γ, there exists an αα-labeling fFfF with fF(V(F))=L∪HfF(V(F))=L∪H, where L,H⊂[0,q]L,H⊂[0,q] are the stable sets induced by the characteristic of fFfF and they do not depend on FF. We also obtain analogous applications to near αα-labelings and bigraceful labelings.