Cycles in 4-connected planar graphs

Let G be a 4-connected planar graph on n vertices. Previous results show that G contains a cycle of length k for each k∈{n,n−1,n−2,n−3} with k≥3. These results are proved using the “Tutte path” technique, and this technique alone cannot be used to obtain further results in this direction. One approach to obtain further results is to combine Tutte paths and contractible edges. In this paper, we demonstrate this approach by showing that G also has a cycle of length k for each k∈{n−4,n−5,n−6} with k≥3. This work was partially motivated by an old conjecture of Malkevitch.