A graph G is called pancyclic if it contains a cycle of length l for each integer l from 3 to |V (G)| inclusive, where |V (G)| denotes the cardinality of the vertex set of graph G. It has been shown by Ma et al. (2007) that the augmented cube, proposed by Choudum and Sunitha (2002), is pancyclic. In this paper, we propose a more refined property, namely double-pancyclicity. Let G be a pancyclic graph with N vertices, and (u1, v1), (u2, v2) be any two vertex-disjoint edges in G. Moreover, let l1 and l2 be any two integers of {3, 4, …, N − 3} such that l1 + l2 ≤ N. Then G is said to be double-pancyclic if it has two vertexdisjoint cycles, C1 and C2, such that |V (Ci)| = li and (ui, vi) ∈ E(Ci) for i = 1, 2. Moreover, we show that the class of augmented cubes can be almost double-pancyclic.
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机译：如果图G包含从3到| V（G）|的每个整数l包含长度为l的循环，则该图称为泛循环。 |包括在内，其中| V（G）|表示图G的顶点集的基数。 （2007年），Choudum和Sunitha（2002年）提出的增广立方是全圈状的。在本文中，我们提出了一种更精细的属性，即双泛环性。令G为具有N个顶点的泛环图，并且（u 1 ，v 1 ），（u 2 ，v 2 < / inf>）是G中任意两个不相交的边。此外，令l 1 和l 2 是{3，4，…，N- 3}，使得l 1 + l 2 ≤N。然后，如果G具有两个顶点不相交的循环C 1 和C 2 ，这样| V（C i ）| = l i 和（u i ，v i ）∈E（C i ）对于i = 1 2.此外，我们证明了增广立方的类别几乎可以是双全环。
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