Long cycles in 4-connected planar graphs

Let G be a 4-connected planar graph on n vertices. Malkevitch conjectured that if G contains a cycle of length 4, then G contains a cycle of length k for every k ∈ { n , n − 1 , … , 3 } . This conjecture is true for every k ∈ { n , n − 1 , … , n − 6 } with k ≥ 3 . In this paper, we prove that G also has a cycle of length n − 7 provided n ≥ 10 .