NON-HAMILTONIAN PLANAR MAPS

Abstract

This chapter discusses the non-Hamiltonian planner maps. A planar map is a dissection of the sphere or closed plane into a finite number of simply connected polygonal regions called faces or countries by means of a graph drawn in the surface. It is assumed that this graph has no loop or isthmus. A Hamiltonian circuit in a map is a circuit in its graph passing through every vertex. A map is called Hamiltonian or non-Hamiltonian according to as it does or does not have such a circuit. A Hamiltonian bond in a graph G is a set H of edges such that the rest of the graph consists of two disjoint trees, and each edge of H has one end in each tree. Denoting the number of vertices of G of valency i by fi, and suppose fi' of these to be in the first tree and fi’ in the second. This form of the theory applies to all graphs, whether planar or nonplanar.