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Prof. Dr. Günter Schaar Born in 1932 at Memel (Klaipeda). Promotion in 1962, Habilitation in 1969. Since 1974 Professor of Mathematics at Bergakademie Freiberg. Fields of interest: Discrete Mathematics, Algebra Born in 1957 at Penig. Since 1978 at Bergakademie Freiberg. Studied Mathematics from 1978 to 1982. Promotion in 1985. Fields of interest: Discrete Mathematics, Algebra Dr. Hanns-Martin Teichert Born in 1954 at Halle. Study of Mathematics 1975 - 1980 and promotion in 1983 at Bergakademie Freiberg. Since 1984 working at the Computer Department of Rostock District Hospital. Fields of interest: Discrete Mathematics, Medical Computer Science and Biomathematics Schaar, Günter: Hamiltonian prc Schaar ; Sonnte 1988. - 148 S. (Teubner-Texte NE: Sonntag, Mi ISBN 3-322-005C ISSN 0138-502X ©BSB B. G. Te\ VLN 294-375/74/ Lektor: Dipl.-r Printed in the Gesamtherstelli Betrieb des Gr. Bestell-Nr.: 6' 01550 TEUBNER-TEXTE zur Mathematik Band 108 Herausgeber / Editors: Beratende Herausgeber /Advisory Editors: Herbert Kurke, Berlin Rüben Ambartzumian, Jerevan Joseph Mecke, Jena David E. Edmunds, Brighton Rüdiger Thiele, Leipzig Alois Kufner, Prag Hans Triebei, Jena Burkhard Monien, Paderborn Gerd Wechsung, Jena Rolf J. Nessel, Aachen Claudio Procesi, Rom Kenji Ueno, Kyoto Günter Schaar Martin Sonntag Hanns-Martin Teichert Hamiltonian Properties of Products of Graphs and Digraphs This book gives a survey on the main results concerning the subject described by the title, also considering the contributions made by the authors in this field* The central object is to study the de pendence of the Hamiltonian behaviour of given products of graphs on properties of the factors. Moreover, the classical products (Car tesian sum, lexicographic product, disjunction, Cartesian product, normal product) are particularly investigated in connection with such Hamiltonian properties as traceability, Hamiltonicity, higher Hamil- tonicity, Hamiltonian connectedness, strong path-connectedness, pan cyclicity, dacomposability into Hamiltonian cycles. The parallel treatment of this set of problems for undirected and directed graphs provides the possibility of a comparative consideration with regard to similarities and differences. 1 Das Buch gibt einen Überblick über die Hauptergebnisse zu dem im Ti tel abgesteckten Themenkreis unter Berücksichtigung der speziell von den Autoren auf diesem Gebiet geleisteten Beiträge* Zentraler Gegen stand ist das Studium der Abhängigkeit des hamiltonschen Verhaltens vorgegebener Graphenprodukte von Eigenschaften der Faktoren. Unter sucht werden dabei insbesondere die klassischen Produktbildungen (Cartesische Summe, lexikographisches Produkt, Disjunktion, Carte- sisches Produkt, normales Produkt) in Verbindung mit solchen hamil tonschen Eigenschaften wie Durchlaufbarkeit, Hamiltonizität, höhere Hamiltonizität, hamiltonscher Zusammenhang, starker Wegzusammenhang, Panzyklizität, Zerlegbarkeit in Hamiltonkreise. Die parallele Behand lung der betreffenden Problematik für ungerichtete und gerichtete Graphen bietet die Möglichkeit vergleichender Betrachtung im Hinblick auf Gemeinsamkeiten und Unterschiede. Le livre présent donne un aperçu sur les résultats principaux dans le domaine des problèmes caractérisés par le titre, en tenant compte des contributions faites jusqu'ici par les auteurs. Le sujet central c'est l’étude de la dépendance du comportement hamiltonien des produits de graphes donnés de certaines propriétés de leurs facteurs. On examine spécialement les produits classiques (somme cartésienne, pr^di'it lexicoqraphique. disjonction, produit cartésien, produit normal) en relation avec telles propriétés hamiltoniennes que traversabilité, hamiltonicité, hamiltonicité supérieure, connexion hamiltonienne, forte connexion des chemins, pancyclicité, décomposabilité en cycles hamiltoniens. Le traitement parallèle des problèmes mentionnés concernant des graphes non dirigés et graphes dirigés rend possible une comparaison des ressemblences et des différences. В книге излагается обзор главных результатов проблематики связанной с ее названием, имея в виду вклад внесенный в эту область авторами. Предметом изучения является исследование зависимости гамильтонова поведения данных произведений графов от свойств факторов. При этом особенно исследуются классические произведения (декартова сумма, лексикографическое произведение, дизъюнкция, декартово произведение, нормальное произведение) в связи с такими гамильтоновыми свойствами, как пробегаемость, гамильтоничность, гамильтоничность высшего порядка, гамильтонова связность, сильная цепная связность, пан цикличность, разложимость на гамильтоновы циклы. Одновременное исучение этой проблематики для неориентированных и ориентированных графов дает возможность сопоставления общего и различия между ними. 2 Preface and introductory remarks 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 prop erties - in the sense of a collective denotation - such properties of graphs being related to the existence of Hamiltonian paths or Hamil tonian 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 respec tive 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 bib liographic survey on products of graphs by Dörfler 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. In order to keep the volume within the bounds suitable to the sub ject of the book and in addition, not to overload and weary the reader with lots of overelaborate distinctions of cases, similar proof schemes and accidental details we have left out a full argu 3 mentation for numerous statements so as to concentrate on the main argument« Consequently« we have sought to demonstrate some typical proof methods exemplarily to highlight the principal ideas« In effect there are three proof techniques applied: The first consists in the direct construction of the required paths and cycles in a given graph- product« thereby inspecting all possible cases that may occur in de pendence on the structure of the factors; the second makes use of special algorithms for generating the wanted paths and cycles step by step« and it is particularly suitable for examining higher Hamiltonian properties of graph-products (r-traceability« r-Hamiltonicity« etc«); the third is characterized by using group-theoretic means and is ap plied to the treatment of products of Cayley digraphs« The distinct consideration of undirected graphs (Part 1« Chapters 1 - 8) and digraphs (Part II« Chapters 9 - 16) is not only a conse quence of the personal division of labour in preparing this book; in our opinion this bipartition is well founded for objective reasons because of the different situations as well as for formal reasons due to the greater lucidity of the presentation« Either of the two parts has a preliminary chapter containing some fundamental concepts and notations needed in that part« and it is closed by a self-contained bibliography« an index of the definitions and a list of the used de notations« Moreover« for the arrangement of the contents we preferred« not in each case but generally« a subdivision according to the Hamil tonian properties to be treated« as can be seen from the titles of Chapters 2«3«4«10«12v13 and of the several sections in Chapters 3 and 10; the consideration of the graph-products we are interested in then takes place within the chapters concerned« partly in self-contained sections« The only exceptions are Chapters 5 and 11* References in the text (especially references to the bibliography) without an ad ditional direction at Part I or Part II are always applied to matters within the part under consideration* The parallel treatment of problems of a rather similar nature for undirected graphs and for digraphs renders it possible to notice communities and differences regarding the Hamiltonian behaviour of thé products considered« In order to make these affairs become clearer foir both types of graphs we have provided notions and properties of the same kind with the same term and the same denotation« and we have used fairly uniform symbolism« We believe« therefore« that there should be no cause for confusion« Finally, we want to thank all those who have supported us in pre paring and writing this booklet; particularly we are indebted to all fellow-specialists from both near and far, whose results we have been able to use, as well as to Mrs* 1* Gugel and Mrs« H. Zimmerraann for careful typewriting of this text. Grateful acknowledgements are also due to the Teubner Publishing House and to the editors of the "Teubner-Texte zur Mathematik" for including our book in this series, and to Dr. Renate Müller for her appreciative cooperation. Freiberg, December 1987 The authors 5 Contents Preface ar.d introductory remarks • • • « • • • • 3 Part I: Hamiltonian properties of products of undirected graphs 7 1« Basic definitions and notations • • • • • • 7 2. Hamiltonian cycles and Hamiltonian paths • • • • ! ! 2.1 The Cartesian sum 2*2 Other products • • 21 3» Generalized Hamiltonian properties • • • • • 31 3*1 Properties related to Hamiltonicity ................... 31 3*2 Pancyclicity ...................................................40 4« Decomposition into edge-disjoint Hamiltonian cycles • 48 4*1 The Cartesian s u m ..........................................48 4.2 Other products..............................................54 5« Generalizations of the classical products • • • 57 6. References ...................................................66 7. Index of definitions • • • • • • • • • 7 0 8« Index of notations..........................................71 Part II: Hamiltonian properties of products of digraphs • • 74 9. Basic definitions and notations • • • • • • 74 10. r-Hamiltonian properties .................................. 7 8 10.1 Traceability ...................................................79 10.2 Homogeneous traceability • • « • • • • • 8 8 10*3 Hamiltonicity...............................................97 10.4 Hamiltonian connectedness ............................. 100 11« Products of Cayley digraphs ............................. 109 11*1 The Cartesian sum .........................................Ill 11.1.1 Directed cycles ........................ • • • • 111 11.1.2 Other Cayley digraphs « • « • • • • • 115 11.2 The Cartesian p r o d u c t .................... • • • 121 12. Strong path-connectedness ............................ 125 13. Pancyclic properties ...................................... 134 14. References............................ 142 15« Index of definitions • • • • • • • • • 144 16. Index of notations • • • • • • « • « 145 5 Part I Hamiltonian properties of products of undirected graphs 1. Basic definitions and notations All graphs considered in Part I of this book are supposed to be undirected, nonempty, simple (i.e. without loops and multiple edges), and - so far as nothing is stated to the contrary - finite. Concerning the terminology we refer to Harary [29j where the reader can find the definitions of the customary graph-theoretic concepts which we shall use without explaining. Furthermore, a great number of additional notions is needed, the most important of them we will introduce in this chapter. Above all we settle some notations. For a graph G let V(G) and E(G) denote the vertex-set of G and the edge-set of G, respectively; we write G = (V(G),E(G))• Edges are written as sets of two distinct vertices, edge-sequences are represented by arranging the passed vertices; more precisely: the edge e£E(G) with the end-vertices x,ye V(G), x $ y, is denoted by fx,y}, and every edge-sequence w in G is written as a sequence w = (xQ»xlf • . • ,xn) with n ^ 0, x±€V(G), i = 0,... ,n, and txi*xi+l^6 E(G)• 1 = The number l(w) := n is the length of w, and if w^ = (xQf...,xk)f wg = (xk+1*•••*xn) for some 0 ^ k < n we use the notations w = (w^.Wg) = • • • *xn^ = = (xQ,...,xk,w2) = (xq,•..,x^,xk+i*•••*xn)* analogously for more than two w's. Further it is useful to define w = (x~,...,x ) := J0T i + ' O n ' if n < 0 (empty sequence). By v(x:G)f v (G) and dG we denote the degree (valency) of the vertex x in the graph G, the maximum degree in G and the distance function of G, respectively. If G is not connected and x,y are vertices belonging to different components of G the distance dG(x,y) is defined to be 00 • A vertex x€V(G) is called an end-vertex of G iff v(x:G) = 1. The graph G' is a sub graph of the graph G (notation: G* £! G) iff V(G* )^ V(G) and • E(G*)SE(G). By G-M with M<=V(G) we denote that subgraph of G arising from G by removing the vertices of M and all edges incident to these vertices; analogously G-x for x 6V(G) is defined. 7 Paths and cycles are comprehended to be special graphs (possibly subgraphs of a graph G); they are usually represented by edge- sequences, and as for the notation we do not differ between paths or cycles as graphs and their representations by corresponding edge- sequences. Especially, (x) means the graph (path of length 0) con sisting of the vertex x. An (x.v)-path in G is a path with the end- vertices x and y. (x€ V(G) is the initial vertex and y€V(G) is the terminal vertex of the corresponding edge-sequence.) We write G^~ G2 iff the graphs G^ and G2 are isomorphic. By Pr, Cr# Kp, Kp s,and Kp we denote respectively, the following graphs which are uniquely determined up to isomorphisms: the path with r vertices, the cycle with r vertices (r^3), the complete graph with r vertices, the complete bipartite graph the one vertex-class of which consists of r and the other of s vertices, and the totally disconnected graph with r vertices (i.e. E(i<r) = 0). Let N, N+, N* and N* be the set of natural numbers {0,1,...}, the set of positive natural numbers, the set N v £«o}, and the set of integers, respectively; as usual for any real number x we define ^xj := max £ i € N* : i £ x}. The graph G with V(G) := ViG^v.. .vV(Gr) and E(G) := E(G^)v• • .vE(Gp) is called the union of the r ^ 1 graphs G1,...,Gf. and denoted by G^vy.^uG^ If x,y are distinct vertices in a graph G the graph arising from G by adding the edge e := {x,y} is denoted by Gue (i.e. Gue := Gv(x,y) where (x,y) is the path of length one with the end-vertices x,y). So far as no other numbering is fixed we will assume that the ver tices of a path Pr or a cycle Cr are denoted in turn along a rep resenting edge-sequence by 0,...,r-l. For the bipartite graph with the vertex-classes V1#V2 and the edge-set E we use the notation (V1#V2;E). A graph G is said to be (i) traceable, (ii) homogeneously traceable, (iii) Hamiltonian, (iv) Hamiltonian-connected, (v) strongly path- connected iff (i) there is a Hamiltonian path in G, (ii) for every x€V(G) there is a Hamiltonian path in G with the initial vertex x, (iii) |V(G)1 ^ 3 and there is a Hamiltonian cycle in G, (iv) for every pair x,y€V(G) of distinct vertices there is a Hamiltonian (x,y)-path in G, (v) for every pair x,y€V(G) of distinct vertices and every 16 N with dG(x,y) ^ 1 ^ |V(G)| - 1 there is an (x,y)-path of length 1 in G, and G is connected. Let TG = Q1, HTG = Q2 and HG = denote the class of all traceable, of all homogeneously traceable and of all Hamiltonian graphs. 8