Table Of ContentProf. 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
