Long cycles in 4-connected planar graphs

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