Non-Hamiltonian non-Grinbergian graphs

Abstract Settling a question of Tutte and a similar question of Grunbaum and Zaks, we present a 3-valent 3-connected planar graph that has only pentagons and octagons, has 92 (200, respectively) vertices and its longest circuit (path, respectively) contains at most 90 (198, respectively) vertices; moreover, it is shown that the family of all 3-valent 3-connected planar graphs, having n -gons only if n ≡ 2 (mod3) (or n ≡ 1 (mod3)), has a shortness exponent which is less than one.