Large cycles in 4-connected graphs

The first result states that every 4-connected graph G with minimum degree δ and connectivity κ either contains a cycle of length at least 4 δ − κ − 4 or every longest cycle in G is a dominating cycle. The second result states that any such graph under the additional condition δ ≥ α either contains a cycle of length at least 4 δ − κ − 4 or is hamiltonian, where α denotes the independence number of G . Both results are sharp in all respects.