Pancyclicity of Hamiltonian and highly connected graphs

A celebrated theorem of Chvátal and Erdős says that G is Hamiltonian if κ ( G ) ⩾ α ( G ) , where κ ( G ) denotes the vertex connectivity and α ( 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 κ ( G ) ⩾ 600 α ( 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 δ ( G ) ⩾ 600 α ( G ) then G is pancyclic. Improving an old result of Erdős, we also show that G is pancyclic if it is Hamiltonian and n ⩾ 150 α ( G ) 3 . Our arguments use the following theorem of independent interest on cycle lengths in graphs: if δ ( G ) ⩾ 300 α ( G ) then G contains a cycle of length ℓ for all 3 ⩽ ℓ ⩽ δ ( G ) / 81 .