Some recent results in hamiltonian graphs

Abstract

A variety of recent developments in hamiltonian theory are reviewed. In particular, several sufficient conditions for a graph to be hamiltonian, certain hamiltonian properties of line graphs, and various hamiltonian properties of powers of graphs are discussed. Furthermore, the concept of an n-distant hamiltonian graph is introduced and several theorems involving this special class of hamiltonian graphs are presented.A graph G is defined to be hamiltonian if it has a cycle containing all the vertices of G; such a cycle is called a hamiltonian cycle. The basic unsolved problem of determining which graphs are hamiltonian has led to the investigation of many related problems and, consequently, to the development of what might generally be termed "hamiltonian theory" (see [12,15, 421). In this article we consider a sample of some of the more recent results in the theory of hamiltonian graphs and provide the reader with a large number of references to related results. Because of the vast amount of research taking place in this area of graph theory, several important topics in hamiltonian theory have necessarily been omitted. Among these are hypohamiltonian graphs, the existence of cycles or paths in graphs, the decomposition of graphs into hamiltonian cycles, and hamiltonian directed graphs. We have restricted our attention primarily to results involving sufficient conditions for a graph to be hamiltonian and various hamiltonian properties of line graphs and powers of graphs.In view of the fact that no genuinely useful characterization of hamiltonian graphs exists or seems likely to be found, a good deal of effort has been devoted to developing sufficient conditions for a graph to be hamiltonian. Chronologically, Dirac [22], Ore [48], Pbsa [5 11, Bondy [5], and Chvital[14] have determined such conditions in terms of the degrees of the vertices of a graph, with each successive result strengthening those preceding it.

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