Submaps of maps. II. Cyclically k-connected planar cubic maps

Abstract We show that, for 1≤ k ≤5, if a map M can be contained in a k -cycle so as to preserve k -connectivity, then almost all cyclically k -connected cubic graphs with n vertices contain at least cn copies of M as submaps for some constant c ( M ). By using Walther's construction for maps without Hamiltonian cycles, we obtain an M with which we prove that almost no cyclically k -connected cubic maps with n vertices have a path of length greater than cn , where c is a constant.