On Longest Cycles in Essentially 4-connected Planar Graphs

A planar 3-connected graph G is essentially 4-connected if for any 3-separator S of G, one component of the graph obtained from G by removing S is a single vertex. B. Jackson and N.C. Wormald proved that an essentially 4-connected planar graph on n vertices contains a cycle C such that |V (C)| ≥ 2n+4 5 . For a cubic essentially 4-connected planar graph G, B. Grünbaum, J. Malkevitch , and C.-Q. Zhang showed that G has a cycle on at least 3 4 n vertices. The result of B. Jackson and N.C. Wormald is improved. Moreover, new lower bounds on the length of a longest cycle of G are presented if G is an essentially 4-connected planar graph of maximum degree 4 or G is an essentially 4-connected maximal planar graph.