Abstract
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 $\Gamma$ be a family of digraphs such that $V(F)=V$ for every $F\in \Gamma$. Consider any function $h:E(D) \longrightarrow \Gamma$. Then the product $D \otimes_h \Gamma$ is the digraph with vertex set $V(D) \times V$ and $((a,x),(b,y)) \in E(D \otimes_h \Gamma)$ if and only if $(a,b) \in E(D)$ and $(x,y) \in E(h (a,b))$. In this paper we use rainbow eulerian multidigraphs and permutations as a way to characterize the $\otimes_h$-product of oriented cycles. We study the behavior of the $\otimes_h$-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.
Highlights
For Cn−the the undefined concepts appearing in two possible strong orientations this paper, we refer the of the cycle Cn and by r→−Geadaenrytoor[i1e7n]t.atWioen denote by Cn+ of a graph G.and Let by D be a digraph, we denote by D− the reverse of D, that is, the digraph obtained from D by reversing all its arcs
We say that M is rainbow eulerian if it has an eulerian circuit in which a sequence of l colors repeats
We study Cm+ ⊗h Γ, where Γ is a family of 1-regular digraphs. We characterize this product in terms of rainbow eulerian multidigraphs (Theorem 2.2), which leads us to a further characterization in terms of permutations (Theorem 2.6)
Summary
The the undefined concepts appearing in two possible strong orientations this paper, we refer the of the cycle Cn and by r→−Geadaenrytoor[i1e7n]t.atWioen denote by Cn+ of a graph G. Let by D be a digraph, we denote by D− the reverse of D, that is, the digraph obtained from D by reversing all its arcs According to this notation, it is clear that (Cn+)− = Cn−. Let M be an arc labeled eulerian multidigraph with l colors. We study Cm+ ⊗h Γ, where Γ is a family of 1-regular digraphs We characterize this product in terms of rainbow eulerian multidigraphs (Theorem 2.2), which leads us to a further characterization in terms of permutations (Theorem 2.6). This is the content of Section 2. We construct families of graphs with an increasing number of possible magic-sums (Theorem 4.4)
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