A note on 3-connected cubic planar graphs

The length of a longest cycle in a graph G is called the circumference of G and is denoted by c ( G ) . Let c ( n ) = min { c ( G ) : G  is a 3-connected cubic planar graph of order  n } . Tait conjectured in 1884 that c ( n ) = n , and Tutte disproved this in 1946 by showing that c ( n ) ≤ n − 1 for n = 46 . We prove that the inequality c ( n ) ≤ n − n + 49 4 + 5 2 holds for infinitely many integers n . The exact value of c ( n ) is unknown.