Longest Cycles in 3-connected 3-regular Graphs

I n t r o d u c t i o n . In this paper, we s tudy the following quest ion: How long a cycle must there be in a 3-connected 3-regular graph on n vertices? For planar graphs this question goes back to T a i t [6], who conjectured tha t any planar 3-connected 3-regular graph is hamiltonian. T u t t e [7] disproved this conjecture by finding a counterexample on 46 vertices. Using Tu t t e ' s example, Grunbaum and Motzkin [3] constructed an infinite family of 3-connected 3-regular planar graphs such tha t the length of a longest cycle in each member of the family is a t most n, where c = 1 — 2~ and n is the number of vertices. The exponent c was subsequently reduced by Walther [8, 9] and by Grùnbaum and Walther [4]. It is natural to ask what one can say when the planari ty condition is dropped. For 2-connected 3-regular graphs, Bondy and Entr inger [2] proved tha t the length of a longest cycle is a t least 4 log2?z — 4 log2log2w — 20, and an example due to Lang and Walther [5] shows tha t this result is essentially best possible. Let f(n) denote the largest integer k such tha t every 3-connected 3-regular graph on n vertices contains a cycle of length a t least k. For planar graphs, Barnet te [1] proved tha t