A new application of the $\otimes_h$-product to $\alpha$-labelings

TL;DR

This paper generalizes the weak tensor product for constructing graphs with $ ext{alpha}$-labelings by incorporating a family of bipartite graphs, enabling new applications to near and bigraceful labelings.

Contribution

It introduces a generalized $ ext{h}$-product involving a family of bipartite graphs for constructing $ ext{alpha}$-labelings, extending previous methods.

Findings

Generalized the $ ext{h}$-product for $ ext{alpha}$-labelings.

Established applications to near $ ext{alpha}$-labelings.

Extended to bigraceful labelings.

Abstract

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 $\Gamma$ of bipartite $(p,q)$-graphs, $p$ and $q$ fixed. The only additional restriction that we should consider is that for every $F\in \Gamma$, there exists and $\alpha$-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.