Non-Hamiltonian simple 3-polytopes having just two types of faces

AbstractIt is shown that for every value of an integer k, k⩾11, there exist 3-valent 3-connected planar graphs having just two types of faces, pentagons and k-gons, and which are non- Hamiltonian. Moreover, there exist ϵ=ϵ(k) > 0, for these values of k, and sequences (Gn)∞n=1 of the said graphs for which V(Gn)→∞ and the size of a largest circuit of Gn is at most (1−ϵ)V(Gn); similar result for the size of a largest path in such graphs is established for all k, k⩾11, except possibly for k = 14, 17, 22 and k = 5m+ 5 for all m⩾2