  An alternating cycle in a 2-two-edge-colored graph is a cycle such that any
two consecutive edges have different colors. Let $G_1, \ldots, G_k$ be a
collection of pairwise vertex disjoint 2-edge-colored graphs. The colored
generalized sum of $G_1, \ldots, G_k$, denoted by $ \oplus_{i=1}^k G_i$, is the
set of all 2-edge-colored graphs $G$ such that: (i) $V(G)=\bigcup_{i=1}^k
V(G_i)$, (ii) $G\langle V(G_i)\rangle\cong G_i$ for $i=1,\ldots, k$ as
edge-colored graphs where $G\langle V(G_i)\rangle$ has the same coloring as
$G_i$ and (iii) between each pair of vertices in different summands of $G$
there is exactly one edge, with an arbitrary but fixed color. A graph $G$ in
$\oplus_{i=1}^k G_i$ will be called a colored generalized sum (c.g.s.) and we
will say that $e\in E(G)$ is an exterior edge iff $e\in E(G)\setminus
\left(\bigcup_{i=1}^k E(G_i)\right)$. The set of exterior edges will be denoted
by $E_\oplus$. A colored graph $G$ is said to be a vertex alternating-pancyclic
graph, whenever for each vertex $v$ in $G$, and for each $l\in\{3,\ldots,
|V(G)|\}$, there exists in $G$ an alternating cycle of length $l$ passing
through $v$.
  The topics of pancyclism and vertex-pancyclism are deeply and widely studied
by several authors. The existence of alternating cycles in 2-edge-colored
graphs has been studied because of its many applications. In this paper, we
give sufficient conditions for a graph $G\in \oplus_{i=1}^k G_i$ to be a vertex
alternating-pancyclic graph.
