Pancyclicity of Hamiltonian and highly connected graphs

A celebrated theorem of Chvatal and Erdos says that G is Hamiltonian if kappa(G) >= alpha(G), where kappa(G) denotes the vertex connectivity and alpha(G) the independence number of G. Moreover, Bondy suggested that almost any non-trivial conditions for Hamiltonicity of a graph should also imply pancyclicity. Motivated by this, we prove that if kappa(G) >= 600 alpha(G) then G is pancyclic. This establishes a conjecture of Jackson and Ordaz up to a constant factor. Moreover, we obtain the more general result that if G is Hamiltonian with minimum degree delta(G) >= 600 alpha(G) then G is pancyclic. Improving an old result of Erdos, we also show that G is pancyclic if it is Hamiltonian and n >= 150 alpha(G)(3). Our arguments use the following theorem of independent interest on cycle lengths in graphs: if delta(G) >= 300 alpha(G) then G contains a cycle of length l for all 3 <= l <= delta(G)/81. (C) 2010 Elsevier Inc. All rights reserved.