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. Jackson and Wormald proved that an essentially 4 connected planar graph on n vertices contains a cycle C such that vertical bar V(C)vertical bar >= 2n+4/5 For a cubic essentially 4-connected planar graph G, Grunbaum with Malkevitch, and Zhang showed that G has a cycle on at least 3/4 n vertices. In the present paper the result of Jackson and 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.