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 - root n + 49/4 + 5/2 holds for infinitely many integers n. The exact value of c(n) is unknown. Published by Elsevier B.V.