Geodesic pancyclicity and balanced pancyclicity of Augmented cubes

Abstract

For two distinct vertices u , v ∈ 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 ) | x , y ∈ 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 | V ( G ) | 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 u , v ∈ V ( G ) , it contains a geodesic cycle (respectively, balanced cycle) of every integer length of l satisfying max { 2 d G ( u , v ) , 3 } ⩽ l ⩽ | V ( G ) | . Lai et al. [P.L. Lai, J.W. Hsue, J.J.M. Tan, L.H. Hsu, On the panconnected properties of the Augmented cubes, in: Proceedings of the 2004 International Computer Symposium, 2004, pp. 1249–1251] proved that the n-dimensional Augmented cube, AQ n , is pancyclic in the sense that a cycle of length l exists, 3 ⩽ l ⩽ | V ( AQ n ) | . In this paper, we study two new pancyclic properties and show that AQ n is geodesic pancyclic and balanced pancyclic for n ⩾ 2 .