Longest Cycles in 3-Connected 3-Regular Graphs

Introduction. In this paper, we study the following question: 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 Tait [6], who conjectured that any planar 3-connected 3-regular graph is hamiltonian. Tutte [7] disproved this conjecture by finding a counterexample on 46 vertices. Using Tutte's example, Grunbaum and Motzkin [3] constructed an infinite family of 3-connected 3-regular planar graphs such that the length of a longest cycle in each member of the family is at most nc, where c = 1 — 2~17 and n is the number of vertices. The exponent c was sub­ sequently reduced by Walther [8, 9] and by Grunbaum and Walther [4]. It is natural to ask what one can say when the planarity condition is dropped. For 2-connected 3-regular graphs, Bondy and Entringer [2] proved that the length of a longest cycle is at least 4 log 2?z — 4 log 2log2w — 20, and an example due to Lang and Walther [5] shows that this result is essentially best possible. Let f(n) denote the largest integer k such that every 3-connected 3-regular graph on n vertices contains a cycle of length at least k. For planar graphs, Barnette [1] proved that