Twisted cubes, crossed cubes, Möbius cubes, and locally twisted cubes are some of the widely studied hypercube variants. The 4-pancyclicity of twisted cubes, crossed cubes, Möbius cubes, locally twisted cubes and the 4-edgepancyclicity of crossed cubes are proven in [2, 1, 5, 10, 6] respectively. It should be noted that 4-edge-pancyclicity implies 4-node-pancyclicity which further implies 4-pan...

a r t i c l e i n f o a b s t r a c t The plausibility of embedding cycles of different lengths in the graphs of a network (known as the pancyclicity property) has important applications in interconnection networks, parallel processing systems, and the implementation of a number of either computational or graph problems such as those used for finding storage schemes of logical data structures, ...

We survey results on paths, trees and cycles in tournaments. The main subjects are hamiltonian paths and cycles, vertex and arc disjoint paths with prescribed endvertices, arc-pancyclicity, oriented paths, trees and cycles in tournaments. Several unsolved problems are included.

Jackson and Ordaz conjectured that if the stability number (G) of a graph G is no greater than its connectivity (G) then G is pancyclic. Applying Ramsey’s theorem we prove the pancyclicity of every graph G with (G) su7ciently large with respect to (G). c © 2003 Elsevier B.V. All rights reserved. MSC: 05C38; 05C45

In the class of k-connected claw-free graphs, we study the stability of some hamilto-nian properties under a closure operation introduced by the third author. We prove that (i) the properties of pancyclicity, vertex pancyclicity and cycle extendability are not stable for any k (i.e., for any of these properties there is an innnite family of graphs G k of arbitrarily high connectivity k such tha...