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Schaar G., Sonntag M., Teiche H.-M. Hamiltonian Properties of Products of Graphs and Digraphs

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Schaar G., Sonntag M., Teiche H.-M. Hamiltonian Properties of Products of Graphs and Digraphs
Teubner, 1988. — 152.
This book aims to offer a survey of the most important results and ideas concerning the field of the Hamiltonian properties of products of undirected and directed graphs* We understand by Hamiltonian properties - in the sense of a collective denotation - such properties of graphs being related to the existence of Hamiltonian paths or Hamiltonian cycles in some way or other, as for instance traceability, Hamiltonicity, Hamiltonian connectedness, pancyclicity. With the only exception of the operation join all the products of graphs considered here are of the type that the vertex-set of the product of graphs is always the Cartesian product of the vertex-sets of the factors given; above all, the investigations are dealing with the five classical products: Cartesian sum, lexicographic product, disjunction, Cartesian product and normal product. The main object of this book, therefore, is to study the dependence of the Hamiltonian behaviour of the respective graph-product on the properties of its factors.
For undirected graphs in the beginning of this decade a relatively abundant literature relating to our subject already existed - the papers concerned which appeared up to 1980 are contained in the bibliographic survey on products of graphs by Dorfler and Music -, whereas in the case of directed graphs (digraphs) until recently there were merely a few relevant publications and, moreover, they dealt almost without exception with the products of Cayley digraphs. Meanwhile, in both directions, the number of results has increased considerably, including the contributions present authors* own, so that to attempt a first synthesising and unifying description can be regarded as completely justified. Actually, the outcome of suchlike efforts presented in this book in some essential parts is based on the theses of the two younger ones in our author-team; however, we hope that we have succeeded in integrating the most important and interesting research results obtained in this field and which have become known to us till the middle of 1986. Of course, we could not avoid making some selection. In case something or other has escaped our notice or failed to be sufficiently considered by us, we would kindly ask for the indulgence of our fellow-specialists.
Part I: Hamiltonian properties of products of undirected graphs
Basic definitions and notations
Hamiltonian cycles and Hamiltonian paths
Generalized Hamiltonian properties
Decomposition into edge-disjoint Hamiltonian cycles
Generalizations of the classical products
Part II: Hamiltonian properties of products of digraphs
Basic definitions and notations
r-Hamiltonian properties
Products of Cayley digraphs
The Cartesian product
Strong path-connectedness
Pancyclic properties
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