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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 , who conjectured that any planar 3-connected 3-regular graph is hamiltonian. Tutte  disproved this conjecture by finding a counterexample on 46 vertices. Using Tutte's example, Grunbaum and Motzkin  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 subsequently reduced by Walther [8, 9] and by Grùnbaum and Walther .
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