Table Of ContentDenes Konig
Theory of Finite
and Infinite Graphs
Translated by Richard McCoart
With Commentary by W.T. Tutte
With 112 Illustrations
Birkhauser
Boston . Basel . Berlin
Translator
Richard McCoart
Department of Mathematics
Loyola College
Baltimore, Maryland 21210-2699
USA
Author of commentary
W.T. Tutte
Department of Mathematics
University of Waterloo
Waterloo, Ontario
Canada N2L 3G 1
Originally published as "Theorie der endlichen und unendlichen Graphen"
Akademische Verlagsgesellschaft Leipzig 1936. German edition © 1986
by B.G. Teubner GmbH.
Library of Congress Cataloging-in-Publication Data
Konig, D. (Denes), 1884-1944.
[Theorie der endlichen und unendlichen Graphen. English]
Theory of finite and infinite graphs 1 Denes Konig ; translated by
Richard McCoart ; with commentary by W.T. Tutte.
p. cm.
Translation of: Theorie der endlichen und unendlichen Graphen.
Includes bibliographical references.
ISBN 978-1-4684-8973-6
I. Graph theory. 2. Konig, D., (Denes), 1884-1944. I. Title.
QAI66.K6613 1989
511' .5-dc20 89-39380
Printed on acid-free paper.
© Birkhiiuser Boston, 1990.
Softcover reprint of the hardcover 1s t edition 1990
All rights reserved. No part of this publication may be reproduced, stored in a retrieval system,
or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or
otherwise, without prior permission of the copyright owner.
ISBN 978-1-4684-8973-6 ISBN 978-1-4684-8971-2 (eBook)
DOl 10.1007/978-1-4684-8971-2
Camera-ready copy provided by the translator.
9 8 7 6 5 432 I
Contents
Commentary
W.T. Tutte ....................................................... .
Theory of Finite and Infinite Graphs
D. Konig . ....... .... ... ... ... ..... ...... ...... .... ................ 45
Denes Konig: A Biographical Sketch
T. Gallai ........................................................... 423
Commentary
1. Introduction.
To most graph theorists there are two outstanding landmarks in
the history of their subject. One is Euler's solution of the Konigsberg
Bridges Problem, dated 1736, and the other is the appearance of
Denes Konig's textbook in 1936. "From Konigsberg to Konig's
book" sings the poetess, "So runs the graphic tale ... " [10].
There were earlier books that took note of graph theory. Veb
len's Analysis Situs, published in 1931, is about general combinato
rial topology. But its first two chapters, on "Linear graphs" and
"Two-Dimensional Complexes", are almost exclusively concerned
with the territory still explored by graph theorists. Rouse Ball's
Mathematical Recreations and Essays told, usually without proofs,
of the major graph-theoretical advances ofthe nineteenth century, of
the Five Colour Theorem, of Petersen's Theorem on I-factors, and of
Cayley's enumerations of trees. It was Rouse Ball's book that kindled
my own graph-theoretical enthusiasm. The graph-theoretical papers
of Hassler Whitney, published in 1931-1933, would have made an
excellent textbook in English had they been collected and published
as such. But the honour of presenting Graph Theory to the mathe
matical world as a subject in its own right, with its own textbook,
belongs to Denes Konig.
Low was the prestige of Graph Theory in the Dirty Thirties.
It is still remembered, with resentment now shading into amuse
ment, how one mathematician scorned it as "The slums of Topol
ogy". It was the so-called science of trivial and amusing problems
for children, problems about drawing a geometrical figure in a single
sweep of the pencil, problems about threading mazes, and problems
about colouring maps and cubes in cute and crazy ways. It was too
hastily assumed that the mathematics of amusing problems must be
trivial, and that if noticed at all it need not be rigorously estab
lished. Students tempted by Graph Theory would be advised by
their supervisors to turn to something respectable or even useful,
2 W.T. TUTTE
like differential equations. I am reminded that my own most re
cent research in Graph Theory has involved differential equations.
Mathematics is One, after all.
There were other fads of that sort. Interest in the infant science
of Astronautics was popularly regarded as a sign of insanity. Nuclear
Physics was disparaged as being unlikely to have any significant im
pact on practical affairs. So Denes Konig did not just introduce
Graph Theory. He strove to lift it out of a Slough of Despond and
to set it upon a height. I would say that he, and those his book
inspired, ultimately succeeded in this task, even though some undue
scorn of Graph Theory persisted into the fifties.
2. Konig's First Chapter.
We find in Konig's first Chapter a proper axiomatization of the
subject, an axiomatization that has required little change in the suc
ceeding fifty years. Alas, he did not succeed in standardizing the ter
minology. This was partly because he wrote in German, a language
that was about to decline as a medium of mathematical communica
tion. So today one school of graph theorists speaks of "vertices" and
"edges" , while another prefers to describe "points" and "lines". And
those coming to the subject from general Topology may still speak
of "O-cells" and "l-cells". Nor are the "nodes" and "branches" of an
earlier time completely forgotten. Wherever a student goes in Graph
Theory he is likely to find competing terminologies, among which he
must choose according to his own preference and convenience.
3. Konig's Second Chapter.
Konig next discusses Eulerian trails and Hamiltonian circuits.
The Eulerian problem is the following. When can we trace a path
in a graph that traverses each edge once and once only, and when
can we arrange for such a path to return to its starting point? The
problem is old; we recognize it in Euler's puzzle of the Seven Bridges,
and we hail his work on the problem as the origin of Graph Theory.
Readers who wish to know what Euler actually proved, and who
shrink from the original Latin, would do well to consult the historical
textbook Graph Theory, 1736-1936 by Biggs, Lloyd and Wilson, [5]
COMMENTARY 3
and to read Robin Wilson's article in the commemorative issue of
the Journal of Graph Theory [52].
The solution of our problem is easily stated. We can construct
the required path if and only if the graph is connected and has at
most two odd vertices. If there are two odd vertices then the path
goes from one of them to the other. In the alternative case there is
no odd vertex at all, and the path terminates at its starting point.
Konig noted that the Euler Problem for infinite graphs "of finite
degree" was still unsolved. But a solution was given by P. Erdos, G.
Grunwald, and E. Vazsonyi in 1938 [12]. Here we may notice some
more examples of the chaotic state of graph-theoretical terminology.
By the "degree" of a vertex Konig means the number of incident
edges, loops if allowed being counted twice. Many graph theorists
still use the term "degree" for this number. But others, including the
present writer, prefer to use "valence" or "valency" for the sake of
the chemical analogy. By an infinite graph "of finite degree" Konig
means a graph in which the valency of each vertex is finite. To some
readers the expression suggests that the valencies of the vertices have
a finite upper bound, but this is not intended. Some of us try to avoid
the suggestion by speaking instead of a "locally finite" infinite graph.
In the third section of the chapter Konig writes about Hamil
tonian circuits. A path in a graph is called Hamiltonian if it visits
each vertex once and once only, except that it may finally return
to its starting point to complete a closed circuit. The Eulerian and
Hamiltonian problems are superficially similar. In the first we de
sire to traverse each edge just once; in the other each vertex. Yet
the second problem is far more difficult than the first. There is as
yet no simple and useable necessary and sufficient condition for the
existence of a Hamiltonian circuit in a given graph.
Konig says he will mention only the two most studied parts of
the theory of these circuits. One is the old problem of the Knight's
tour of the chessboard. There we need a Hamiltonian circuit in the
graph whose vertices are the squares of the board and whose edges
are the possible Knight's moves. In Konig's Figure 12 he shows a
solution due to Euler.
The second part is more general. It is the study of the Hamil
tonian properties of the "cubic" graphs, the graphs in which each
vertex has degree or valency 3. Konig shows a Hamiltonian cubic
4 W.T. TUTTE
graph in Figure 9 and a non-Hamiltonian one in Figure 10. He goes
on to point out that the graph of the regular dodecahedron is Hamil
tonian. Indeed it was Hamilton's interest in the circuits of this figure
that gave "Hamiltonian Circuits" their name.
Historical research [4, 5] suggests that it would be fairer to call
them "Kirkman's Circuits", since T. P. Kirkman had already studied
them as part of his extensive work on the convex polyhedra. More
over in view of Euler's work on Knights' tours they might easily have
been called "Euler paths of the Second Kind".
In the latter half of the nineteenth century the Hamiltonian
circuits of cubic graphs were studied by P. G. Tait. He was, I think,
primarily interested in the Four Colour Problem, and by his time
that puzzle had been reduced to the case of cubic planar graphs.
When such a graph is known to have a Hamiltonian circuit the Four
Colour Problem for the corresponding map becomes trivial. The
faces or "countries" inside the circuit can be coloured alternately
red and blue, and those outside alternately green and yellow. Then
no edge will separate two faces of the same colour. This observation
suggested to Tait that the Four Colour Theorem might be proved by
way of a theory of Hamiltonian circuits of cubic graphs.
Accordingly Tait put forward his famous conjecture about "true
polyhedra". This asserts that if a cubic graph can be represented
as the grid of edges and vertices of a convex polyhedron, then it
must be Hamiltonian. If this Conjecture could be proved then the
Four Colour Theorem would follow as a simple corollary. But Konig
comments that the conjecture is still not proved or disproved at his
time of writing.
We should comment here on the notion of "3-connection" for
cubic graphs. A cubic graph can be called "1-connected" if it is
connected. Such a graph may have what Konig calls a "bridge".
This is an edge whose removal disconnects its graph. The graph
falls apart into two pieces, each connected and each containing one
end of the bridge. The 1-connected cubic graphs with no bridges are
called "2-connected" or "non-separable". A graph with a bridge has,
obviously, no Hamiltonian circuit.
I prefer another term in place of Konig's "bridge" , say ''isthmus''
or "1-bond".
It may be that a 2-connected cubic graph G has a pair of edges
COMMENTARY 5
whose deletion destroys its connection. It falls apart into two con
nected pieces each containing one end of each member of the deleted
pair. Let us call such a pair a "2-bond". A connected cubic graph
having no I-bond or 2-bond is said to be "3-connected".
Now the 3-connected planar cubic graphs correspond to Tait's
"true polyhedra". This is not obvious: it is part of the very difficult
Steinitz Theorem relating graph theory to 3-dimensional geometry
[15, 34]. However to a graph theorist Tait's Conjecture is simply
that every 3-connected planar cubic graph is Hamiltonian.
The non-Hamiltonian planar cubic graphs shown in Konig's Fig
ures 10 and 14 are 2-connected but not 3-connected.
A counterexample to Tait's Conjecture was published [35] in
1946. It appeared together with a proof of Smith's Theorem that
in any cubic graph, planar or otherwise, the number of Hamilto
nian circuits passing through any given edge is even. (The reader is
reminded that zero is an even number). Smith's Theorem has the
curious corollary that if a cubic graph has one Hamiltonian circuit
it must have at least three.
In succeeding years there were attempts to improve upon the
counterexample. They involved the notion of "cyclic n-connection" .
A 3-connected cubic graph may contain a triad of edges whose dele
tion separates it into two connected pieces each containing a cir
cuit. A 3-connected cubic graph having no such triad is "cyclically
4-connected". Such a graph may still have a quartet of edges separat
ing two circuits. But if it does not then it is cyclically 5-connected,
and so on. This writer described a cyclically 4-connected counter
example in 1960 [40], H. Walther described a cyclically 5-connected
one in 1965 [47]. Both were rather complicated. In a sense there
was no need to go further. Euler's Polyhedron Formula ensures that
no planar cubic graph can be cyclically 6-connected, unless we count
some trivial graphs too simple to have separable circuits.
These large counterexamples were put into the shade by the
very simple theory of E. Ya. Grinberg. He showed that it was still
possible to get results in Hamiltonian theory that were both simple
and interesting. Studying his proof one wonders why Kirkman and
Tait never happened upon it [14].
Oddly enough the Grinberg theory, though stated only for cubic
planar graphs, applies equally well to planar graphs in general. The
