All righis reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanic.; 1 , photocopying, recording, or otherwise, without the prior written permission of the publisher Printed in the United States of America. Published simultaneously in Canada. Library of Congress Catalog Card No. 69-19989 To KASIMIR KURAT0WSK1 Who gave K s and K i3 To those who thmujht planarity Was nothing hut topology. K M :MMBttül ma > uam»** PREFACEWhen I was a boy of 14 my father was so ignorant I could hardly stand to have the old man around. But when 1 got to be 21, I was astonished at how much the old man had learned in 7 years. MARK TWAINThere are several reasons for the acceleration of interest in graph theory. It has become fashionable to mention that there are applications of graph theory to some areas of physics, chemistry, communication science, computer technology, electrical and civil engineering, architecture, operational research, genetics, psychology, sociology, economics, anthropology, and linguistics. The theory is also intimately related to many branches of mathematics, including group theory, matrix theory, numerical analysis, probability, topology, and combinatorics. The fact is that graph theory serves as a mathematical model for any system involving a binary relation. Partly because of their diagrammatic representation, graphs have an intuitive and aesthetic appeal. Although there are many results in this field of an elementary nature, there is also an abundance of problems with enough combinatorial subtlety to challenge the most sophisticated mathematician.Earlier versions of this book have been used since 1956 when regular courses on graph theory and combinatorial theory began in the Department of Mathematics at the University of Michigan. It has been found pedagogically advantageous not to in:iude proofs of all theorems. This device has permitted the inclusion of more theorems than would otherwise have been possible. The book can thus be used as a text in the tradition of the "Moore Method." with the student gaining mathematical power by being encouraged to prove all theorems stated without proof. Note, however, that some of the missing proofs are both difficult and long. The reader who masters the content of this book will be qualified to continue with the study of special topics and to apply graph theory iO other fields.An effort has been made to present the various topics in the theory of graphs in a logical ordei, to indicate the historical background, and to clarify the exposition by including figures to illustrate concepts and results. In addition, there are three appendices which provide diagrams of giaphs. VI PREFACEdirected graphs, and trees. The emphasis throughout is on theorems rather than algorithms or applications, v/hich however are occasionally mentioned.There are vast differences in the level of exercises. Those exercises which are neither easy nor straightforward are so indicated by a bold-faced number...

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A list of 31 problems presented here reflects some of the main trends in topological graph theory. INTRODUCTIONDuring the past 10 years or so, about 100 different authors representing a total of about 20 different countries have used recognizably topological methods to obtain graph theoretic results. Although the majority of these authors are best known as graph theorists, many others among them are primarily topologists, algebraists, or computer scientists. The great emphasis on genus in the present collection of problems is about proportional to that in the general literature. Some of the problems are included for their seeming strategic value for obtaining additional results, others because they would strengthen the ties to areas such as group theory or computational complexity, and others mainly due to the longstanding interest of the present authors in their solution.For further information on the context of some of these problems, it may be useful to consult the recent surveys of topological graph theory by Stahl [37] and by White and Beineke [44]. All unexplained graph theoretic terminology here follows the usage of Harary [22].