Some of My Favourite Conjectures: Local Conditions Implying Global Cycle Properties

Abstract

Graph Theory is believed to have begun with the famous Konigsberg Bridge Problem. Figure 1 shows a map of Konigsberg as it appeared in the eighteenth century. The town was spanned by seven bridges passing over the river Pregel and connecting the four land masses on which Konigsberg was built. The townsfolk amused themselves by taking walks through Konigsberg, attempting to cross each of the seven bridges exactly once. In 1736 Euler put an end to their speculation that such a walk did not exist by giving a rigorous argument which proved their conjecture. Motivated by this problem, a graph is called eulerian if it has a closed walk that traverses each edge exactly once. It is well known that a connected graph is eulerian if and only if the degree of each vertex is even. The eulerian problem for connected graphs is thus easily solved by checking whether the degree of each vertex is even. The vertex analogue of eulerian graphs are the Hamiltonian graphs. These are graphs that have a cycle that passes through each vertex exactly once and are named after Sir William Rowan Hamilton who devised the Icosian Game for two players. One of the problems in the game required the first player to select a path of five vertices on the dodecahedron. The second player had to extend this path to a cycle that contained all 20 points of the dodecahedron. If a graph has a cycle that contains all its vertices such a cycle is called a Hamiltonian cycle. In contrast to the eulerian problem, the Hamilton cycle problem, i.e., the problem of determining whether a given graph has a Hamiltonian cycle, has no known simple solution. As a result many sufficient conditions for hamiltonicity have been established. In this chapter we will describe problems and conjectures that have their roots in the Hamilton cycle problem. Open image in new window Fig. 1 A map of the Konigsberg bridges