On the double-pancyclicity of augmented cubes

A graph G is called pancyclic if it contains a cycle of length I for each integer I 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 vertex-disjoint cycles, C₁ and C₂, such that |V(C_i)| = l_i and (u_i, v_i) ∈ E(C_i) for i = 1,2. Moreover, we show that the class of augmented cubes can be almost double-pancyclic.