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)⩾600α(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)⩾600α(G) then G is pancyclic. Improving an old result of Erdős, we also show that G is pancyclic if it is Hamiltonian and n⩾150α3(G). Our arguments use the following theorem of independent interest on cycle lengths in graphs: if δ(G)⩾300α(G) then G contains a cycle of length ℓ for all 3⩽ℓ⩽δ(G)/81.