The power of digraph products applied to labelings

Abstract

The ⊗ h -product was introduced in 2008 by Figueroa-Centeno et al. [15] as a way to construct new families of (super) edge-magic graphs and to prove that some of those families admit an exponential number of (super) edge-magic labelings. In this paper, we extend the use of the product ⊗ h in order to study the well know harmonious, sequential, partitional and ( a , d ) -edge antimagic total labelings. We prove that if a ( p , q ) -digraph with p ≤ q is harmonious and h : E ( D ) ⟶ S n is any function, then u n d ( D ⊗ h S n ) is harmonious. We obtain analogous results for sequential and partitional labelings. We also prove that if G is a (super) ( a , d ) -edge-antimagic total tripartite graph, then n G is (super) ( a ′ , d ) -edge-antimagic total, where n ≥ 3 , and d = 0 , 2 and n is odd, or d = 1 . We finish the paper providing an application of the product ⊗ h to an arithmetic classical result when the function h is constant.