Rainbow eulerian multidigraphs and the product of cycles

Susana López (1) , Francesc-Antoni Muntaner-Batle (2)

1 UPC - Universitat Politècnica de Catalunya = Université polytechnique de Catalogne [Barcelona]
2 UoN - University of Newcastle [Callaghan, Australia]

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.

Mots clés: rainbow eulerian multidigraph; eulerian multidigraph; direct product; $\otimes_h$-product; (super) edge-magic

Domaines: Mathématique discrète [cs.DM]

Susana López, Francesc-Antoni Muntaner-Batle. Rainbow eulerian multidigraphs and the product of cycles. Discrete Mathematics and Theoretical Computer Science, 2016, Vol. 17 no. 3 (3), pp.90-104. ⟨10.46298/dmtcs.2153⟩. ⟨hal-01352847⟩