Abstract An alternating cycle in a 2-edge-colored graph is a cycle such that any two consecutive edges have different colors. Let G 1 , …, G k be a collection of pairwise vertex disjoint 2-edge-colored graphs. The colored generalized sum of G 1 , …, G k , denoted by ⊕ i = 1 k G i , is the set of all 2-edge-colored graphs G such that: (i) V ( G ) = ⋃ i = 1 k V ( G i ) , (ii) G 〈 V ( G i ) 〉 ≅ G i for i = 1 , … , k as edge-colored graphs where G 〈 V ( G i ) 〉 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 ⊕ i = 1 k G i will be called a colored generalized sum (c.g.s.). A 2-edge-colored graph G of order 2 n is vertex alternating-pancyclic iff, for each vertex v ∈ V ( G ) and each k ∈ { 2 , … , n } , G contains an alternating cycle of length 2 k 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 ∈ ⊕ i = 1 k G i to be a vertex alternating-pancyclic graph.

中文翻译：

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2-edge-colored 图的广义和中的顶点交替泛环性

摘要 2边色图中的交替环是任意两条连续边具有不同颜色的环。令 G 1 , …, G k 是成对顶点不相交 2 边彩色图的集合。G 1 , …, G k 的彩色广义和，表示为 ⊕ i = 1 k G i ，是所有 2 边彩色图 G 的集合，使得： (i) V ( G ) = ⋃ i = 1 k V ( G i ) , (ii) G 〈 V ( G i ) 〉 ≅ G i for i = 1 , … , k 作为边着色图，其中 G 〈 V ( G i ) 〉与 G i 具有相同的着色(iii) 在 G 的不同被加数中的每对顶点之间，恰好有一条边，具有任意但固定的颜色。⊕ i = 1 k G i 中的图 G 将称为有色广义和 (cgs)。一个 2 n 阶的 2 边彩色图 G 是顶点交替泛环 iff，对于每个顶点 v ∈ V ( G ) 和每个 k ∈ { 2 , … , n } ，G 包含通过 v 的长度为 2 k 的交替循环。多位作者对泛环论和顶点泛环论的主题进行了深入而广泛的研究。由于其许多应用，已经研究了 2 边彩色图中交替循环的存在。在本文中，我们给出了图 G ∈ ⊕ i = 1 k G i 是顶点交替泛环图的充分条件。