The power of digraph products applied to labelings

The ⊗h⊗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⊗h in order to study the well know harmonious, sequential, partitional and (a,d)(a,d)-edge antimagic total labelings. We prove that if a (p,q)(p,q)-digraph with p≤qp≤q is harmonious and h:E(D)⟶Snh:E(D)⟶Sn is any function, then und(D⊗hSn)und(D⊗hSn) is harmonious. We obtain analogous results for sequential and partitional labelings. We also prove that if GG is a (super) (a,d)(a,d)-edge-antimagic total tripartite graph, then nGnG is (super) (a′,d)(a′,d)-edge-antimagic total, where n≥3n≥3, and d=0,2d=0,2 and nn is odd, or d=1d=1. We finish the paper providing an application of the product ⊗h⊗h to an arithmetic classical result when the function hh is constant.