Vertex-Pancyclism in the Generalized Sum of Digraphs

A digraph $D=(V(D)$, $A(D))$ of order $n\geq 3$ is pancyclic, whenever $D$ contains a directed cycle of length $k$ for each $k\in \{3,\ldots,n\}$; and $D$ is vertex-pancyclic iff, for each vertex $v\in V(D)$ and each $k\in \{3,\ldots,n\}$, $D$ contains a directed cycle of length $k$ passing through $v$. Let $D_1, D_2, \ldots, D_k$ be a collection of pairwise vertex disjoint digraphs. The generalized sum (g.s.) of $D_1, D_2, \ldots, D_k$, denoted by $\oplus_{i=1}^k D_i$ or $D_1\oplus D_2 \oplus \cdots \oplus D_k$, is the set of all digraphs $D$ satisfying: (i) $V(D)=\bigcup_{i=1}^k V(D_i)$, (ii) $D\langle V(D_i) \rangle \cong D_i$ for $i=1,2,\ldots, k$, and (iii) for each pair of vertices belonging to different summands of $D$, there is exactly one arc between them, with an arbitrary but fixed direction. A digraph $D$ in $\oplus_{i=1}^k D_i$ will be called a generalized sum (g.s.) of $D_1, D_2, \ldots, D_k$. Let $D_1, D_2, \ldots, D_k$ be a collection of $k$ pairwise vertex disjoint Hamiltonian digraphs, in this paper we give simple sufficient conditions for a digraph $D\in \oplus_{i=1}^k D_i$ be vertex-pancyclic. This result extends a result obtained by Cordero-Michel, Galeana-S\'anchez and Goldfeder in 2016.