Table Of ContentExercises in Graph Theory
Kluwer Texts in the Mathematical Sciences
VOLUME 19
A Graduate-Level Book Series
The titles published in this series are listed at the end oft his volume.
Exercises
in Graph Theory
by
o.
Melnikov
Department ofM athematics,
Belarus State University,
Minsk, Belarus
v. Sarvanov
Institute ofM athematics,
Belarus Academy of Sciences,
Minsk, Belarus
R. Tysbkevich
V. Yemelichev
and
I. Zverovich
Department ofM athematics,
Belarus State University,
Minsk, Belarus
SPRINGER-SCIENCE+BUSINESS MEDIA, B.V.
A C.I.P. Catalogue record for this book is available from the Library of Congress.
ISBN 978-90-481-4979-7 ISBN 978-94-017-1514-0 (eBook)
DOI 10.l007/978-94-017-1514-0
Printed on acid-free paper
AII Rights Reserved
@1998 Springer Science+Business Media Dordrecht
Originally published by Kluwer Academic Publishers in 1998
No part of the material protected by this copyright notice may be reproduced or
utilized in any form or by any means, electronic or mechanical,
including photocopying, recording or by any information storage and
retrieval system, without written permission from the copyright owner
Contents
Introduction ...... . 1
1 ABC of Graph Theory 3
1.1 Graphs: Basic Notions ........ . 3
1.2 Walks, Paths, Components ..... . 13
1.3 Subgraphs and Hereditary Properties
of Graphs. Reconstructibility . . 20
1.4 Operations on Graphs ..... . 27
1.5 Matrices Associated with Graphs 31
1.6 Automorphism Group of Graph 35
2 Trees 41
2.1 Trees: Basic Notions ..... 41
2.2 Skeletons and Spanning Trees 48
3 Independence and Coverings 55
3.1 Independent Vertex Sets and Cliques 55
3.2 Coverings . . . . 62
3.3 Dominating Sets ....... . 64
3.4 Matchings............ 66
3.5 Matchings in Bipartite Graphs 68
4 Connectivity 71
4.1 Biconnected Graphs and Biconnected Components 71
4.2 k-connectivity .. 75
4.3 Cycles and Cuts 77
5 Matroids 81
5.1 Independence Systems 81
5.2 Matroids.. ... 83
5.3 Binary Matroids 90
6 Planarity 93
6.1 Embeddings of Graphs. Euler Formula. 93
6.2 Plane Triangulation 98
6.3 Planarity Criteria ...... . 99
v
VI CONTENTS
604 Duality and Planarity .. 101
6.5 Measures of Displanarity . 107
7 Graph Traversals 111
7.1 Eulerian Graphs 111
7.2 Hamiltonian Graphs 112
8 Degree Sequences 117
8.1 Graphical Sequences 117
8.2 P-graphical Sequences ... 126
8.3 Split and Threshold Graphs 129
804 Degree Sets and Arity Partitions 132
9 Graph Colorings 135
9.1 Vertex Coloring. 135
9.2 Chromatic Polynomial 141
9.3 Edge Coloring ..... 142
904 Colorings of Planar Graphs 144
9.5 Perfect Graphs ...... . 147
10 Directed Graphs 15-1
10.1 Directed Graphs: Basic Notions. 151
10.2 Reachability and Components .. 154
10.3 Matrices Associated with Digraph 160
lOA Tours and Paths 163
10.5 Tournaments .. 166
10.6 Base and Kernel 167
11 Hypergraphs 173
11.1 Hypergraphs: Basic Notions. 173
11.2 Hypergraph Realizations . . . 179
Answers to Chapter 1: ABC of Graph Theory 183
1.1 Graphs: Basic Notions ........ . 183
1.2 Walks, Paths, Components ..... . 187
1.3 Sub graphs and Hereditary Properties
of Graphs. Reconstructibility . . 192
104 Operations on Graphs . . . . . . 196
1.5 Matrices Associated with Graphs 197
1.6 Automorphism Group of Graph . 201
Answers to Chapter 2: Trees 207
2.1 Trees: Basic Notions ... 207
2.2 Skeletons and Spanning Trees 218
CONTENTS Vll
Answers to Chapter 3: Independence and Coverings 223
3.1 Independent Vertex Sets and Cliques. 223
3.2 Coverings . . . . 231
3.3 Dominating Sets ....... . 233
3.4 Matchings ........... . 234
3.5 Matchings in Bipartite Graphs 237
Answers to Chapter 4: Connectivity 241
4.1 Biconnected Graphs and Biconnected Components 241
4.2 k-connectivity .. 245
4.3 Cycles and Cuts 248
Answers to Chapter 5: Matroids 251
5.1 Independence Systems 251
5.2 Matroids ..... 252
5.3 Binary Matroids 255
Answers to Chapter 6: Planarity 257
6.1 Embeddings of Graphs. Euler Formula. 257
6.2 Plane Triangulation . 262
6.3 Planarity Criteria ..... 263
6.4 Duality and Planarity .. 267
6.5 Measures of Displanarity . 270
Answers to Chapter 7: Graph Traversals 275
7.1 Eulerian Graphs .. 275
7.2 Hamiltonian Graphs .......... . 276
Answers to Chapter 8: Degree Sequences 281
8.1 Graphical Sequences ... . 281
8.2 P-graphical Sequences ..... . 286
8.3 Split and Threshold Graphs .. . 289
8.4 Degree Sets and Arity Partitions 293
Answers to Chapter 9: Graph Colorings 297
9.1 Vertex Coloring. 297
9.2 Chromatic Polynomial .. . 304
9.3 Edge Coloring ....... . 305
9.4 Colorings of Planar Graphs 307
9.5 Perfect Graphs ...... . 309
Answers to Chapter 10: Directed Graphs 313
10.1 Directed Graphs: Basic Notions .. 313
10.2 Reachability and Components ... 314
10.3 Matrices Associated with Digraph 318
10.4 Tours and Paths 320
10.5 Tournaments ........... . 322
viii CONTENTS
10.6 Base and Kernel . 326
Answers to Chapter 11: Hypergraphs 333
11.1 Hypergraphs: Basic Notions. 333
11.2 Hypergraph Realizations . 336
Bibliography 341
Index ... 342
Notations .. 351
Introduction
This book supplements the textbook of the authors" Lectures on Graph The
ory" [6] by more than thousand exercises of varying complexity. The books match
each other in their contents, notations, and terminology. The authors hope that
both students and lecturers will find this book helpful for mastering and verifying
the understanding of the peculiarities of graphs.
The exercises are grouped into eleven chapters and numerous sections accord
ing to the topics of graph theory: paths, cycles, components, subgraphs, re
constructibility, operations on graphs, graphs and matrices, trees, independence,
matchings, coverings, connectivity, matroids, planarity, Eulerian and Hamiltonian
graphs, degree sequences, colorings, digraphs, hypergraphs.
Each section starts with main definitions and brief theoretical discussions.
They constitute a minimal background, just a reminder, for solving the exercises.
Proofs of the presented facts and a more extended exposition may be found in
the mentioned textbook of the authors, as well as in many other books in graph
theory.
Most exercises are supplied with answers and hints. In many cases complete
solutions are given.
At the end of the book you may find the index of terms and the glossary of
notations.
The "Bibliography" list refers only to the books used by the authors during the
preparation of the exercisebook. Clearly, it mentions only a fraction of available
books in graph theory. The invention of the authors was also driven by numerous
journal articles, which are impossible to list here.
The authors are greatly indebted to Professors Eberhard Girlich, Michael
Hazewinkel, Uno Kaljulaid, and Alexander Mikhalev, who have stimulated and
promoted publishing this book in English.
The authors wish to give their credit to Dr. Nikolai Korneenko for improving
the text during the translation from Russian.
O. Melnikov et al., Exercises in Graph Theory
© Springer Science+Business Media Dordrecht 1998
Chapter 1
ABC of Graph Theory
1.1 Graphs: Basic Notions
Let V(2) be the set of all two-elements subsets of a nonempty set V. An ordered
pair (V, E), where E ~ V(2), is called a graph. The elements of V are called the
vertices and the elements of E are called the edges. The sets of vertices and edges
of a graph G will be denoted by VG and EG respectively.
A graph is called finite if the set of its vertices is finite. This book deals with
finite graphs only; therefore in the sequel we shall omit the qualifier "finite".
The number JVGI of vertices of a graph G is called its order and is denoted by
IGI. A graph of order n is called an n-vertex graph.
D 0
--- L
-------
• • • • • • • • •
P2 P3 P4 P5 C C C
3 4 5
a) b)
*
L I2?J
--
•
Kl K2 K3 K4
c) d)
Figure 1.1.1: Examples of graphs
Graphs are conveniently depicted by drawings that consist of points and con
tinuous lines connecting some pairs of the points. The points correspond to the
vertices of a graph and the connecting lines correspond to the edges. Fig. 1.1.1
shows the following graphs: a) the simple paths Pn , n = 2, ... ,5, b) the simple
= =
cycles Cn, n 3, ... ,5, c) the complete graphs Kn, n 1, ... ,4, d) the Pe-
3
O. Melnikov et al., Exercises in Graph Theory
© Springer Science+Business Media Dordrecht 1998
