Geodesic pancyclicity and balanced pancyclicity of Augmented cubes
For two distinct vertices u, v epsilon V(G), a cycle is called geodesic cycle with u and v if a shortest path of G joining u and v lies on the cycle; and a cycle C is called balanced cycle with u and v if d(C) (u, v) = max{d(C) (x, y) vertical bar x, y is an element of V (C)}. A graph G is pancyclic [J. Mitchem, E. Schmeichel, Pancyclic and bipancyclic graphs a survey, Graphs and applications (1982) 271-278] if it contains a cycle of every length from 3 to vertical bar V(G)vertical bar inclusive. A graph G is called geodesic pancyclic [H.C. Chan, J.M. Chang, Y.L. Wang, S.J. Horng, Geodesic-pancyclic graphs, in: Proceedings of the 23rd Workshop on Combinatorial Mathematics and Computation Theory, 2006, pp. 181-187] (respectively, balanced pancyclic) if for each pair of vertices it, U E V (G), it contains a geodesic cycle (respectively, balanced cycle) of every integer length of l satisfying max{2d(G) (u, v), 3}  = 2. (c) 2006 Elsevier B.V. All rights reserved.
geodesic pancyclic pancyclic balanced pancyclic augmented cubes panconnected interconnection networks