On the number of cycles in 3-connected cubic graphs

Let f(n) be the minimum number of cycles present in a 3-connected cubic graph on n vertices. In 1986, C. A. Barefoot, L. Clark, and R. Entringer (Congr. Numer. 53, 1986) showed that f(n) is subexponential and conjectured that f(n) is superpolynomial. We verify this by showing that, for n sufficiently large, 2(n0.17) < f(n) 2(n0.95). (C) 1997 Academic Press.