Rainbow eulerian multidigraphs and the product of cycles

An arc colored eulerian multidigraph with l colors is rainbow eulerian if there is an eulerian circuit in which a sequence of l colors repeats. The digraph product that refers the title was introduced by Figueroa-Centeno et al. as follows: let D be a digraph and let Γ be a family of digraphs such that V (F ) = V for every F ∈ Γ. Consider any function h : E(D) −→ Γ. Then the product D⊗hΓ is the digraph with vertex set V (D)×V and ((a, x), (b, y)) ∈ E(D⊗hΓ) if and only if (a, b) ∈ E(D) and (x, y) ∈ E(h(a, b)). In this paper we use rainbow eulerian multidigraphs and permutations as a way to characterize the ⊗h-product of oriented cycles. We study the behavior of the ⊗h-product when applied to digraphs with unicyclic components. The results obtained allow us to get edge-magic labelings of graphs formed by the union of unicyclic components and with different magic sums.