In graph theory, the related problems of deciding when a set of vertices or a set of edges constitutes a maximum matching or a minimum covering have been extensively studied. In this paper we generalize these ideas by defining total matchings and total coverings, and show that these sets, whose elements in general consist of both vertices and edges, provide a way to unify these concepts. Parameters denoting the maximum and the minimum cardinality of these sets are introduced and upper and lower bounds depending only on the order of the graph are obtained for the number of elements in arbitrary total matchings and total coverings. Precise values of all the parameters are found for several general classes of graphs, and these are used to establish the sharpness of most of the bounds. In addition, variations of some well known equalities due to Gallai relating covering and matching numbers are obtained.
A set E of edges of a graph G is said to be a dominating set of edges if every edge of G either belongs to E or is adjacent to an edge of E. If the subgraph 〈E〉 induced by E is a trail T, then T is called a dominating trail of G. Dominating circuits are defined analogously. A sufficient condition is given for a graph to possess a spanning (and thus dominating) circuit and a sufficient condition is given for a graph to possess a spanning (and thus dominating) trail between each pair of distinct vertices. The line graph L(G) of a graph G is defined to be that graph whose vertex set can be put in one-to-one correspondence with the edge set of G in such a way that two vertices of L(G) are adjacent if and only if the corresponding edges of G are adjacent. The existence of dominating trails and circuits is employed to present results on line graphs and second iterated line graphs, respectively.
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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