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Graph Theory Graph Theory (II) J.A. Bondy U.S.R. Murty Graph Theory (II) ABC J.A. Bondy, PhD U.S.R. Murty, PhD Universite´ Claude-Bernard Lyon 1 Mathematics Faculty Domaine de Gerland University of Waterloo 50 Avenue Tony Garnier 200 University Avenue West 69366 Lyon Cedex 07 Waterloo, Ontario, Canada France N2L 3G1 Editorial Board S. Axler K.A. Ribet Mathematics Department Mathematics Department San Francisco State University University of California, Berkeley San Francisco, CA 94132 Berkeley, CA 94720-3840 USA USA Graduate Texts in Mathematics series ISSN: 0072-5285 ISBN: 978-1-84628-969-9 e-ISBN: 978-1-84628-970-5 DOI: 10.1007/978-1-84628-970-5 Library of Congress Control Number: 2007940370 Mathematics Subject Classification (2000): 05C; 68R10 °c J.A. Bondy & U.S.R. Murty 2008 Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or trans- mitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms of licenses issued by the Copyright Licensing Agency. Enquiries concerning reproduction outside those terms should be sent to the publishers. The use of registered name, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant laws and regulations and therefore free for general use. The publisher makes no representation, express or implied, with regard to the accuracy of the information contained in this book and cannot accept any legal responsibility or liability for any errors or omissions that may be made. Printed on acid-free paper 9 8 7 6 5 4 3 2 1 springer.com Contents 1 Graphs ........................................................ 1 2 Subgraphs ..................................................... 39 3 Connected Graphs ............................................. 79 4 Trees .......................................................... 99 5 Nonseparable Graphs ..........................................117 6 Tree-Search Algorithms ........................................135 7 Flows in Networks .............................................157 8 Complexity of Algorithms .....................................173 9 Connectivity ...................................................205 10 Planar Graphs .................................................243 11 The Four-Colour Problem .....................................287 12 Stable Sets and Cliques ........................................295 13 The Probabilistic Method .....................................329 14 Vertex Colourings .............................................357 15 Colourings of Maps ............................................391 16 Matchings .....................................................413 17 Edge Colourings ...............................................451 XII Contents 18 Hamilton Cycles ...............................................471 19 Coverings and Packings in Directed Graphs ...................503 20 Electrical Networks ............................................527 21 Integer Flows and Coverings ...................................557 Unsolved Problems ................................................583 References .........................................................593 General Mathematical Notation ...................................623 Graph Parameters .................................................625 Operations and Relations ..........................................627 Families of Graphs .................................................629 Structures .........................................................631 Other Notation ....................................................633 Index ..............................................................637 10 Planar Graphs Contents 10.1 Plane and Planar Graphs .........................243 The Jordan Curve Theorem ......................... 244 Subdivisions ......................................... 246 10.2 Duality ..........................................249 Faces ................................................ 249 Duals ............................................... 252 Deletion–Contraction Duality ...................... 254 Vector Spaces and Duality .......................... 256 10.3 Euler’s Formula ..................................259 10.4 Bridges ..........................................263 Bridges of Cycles ................................... 264 Unique Plane Embeddings ............................ 266 10.5 Kuratowski’s Theorem ............................268 Minors .............................................. 268 Wagner’s Theorem ................................... 269 Recognizing Planar Graphs ......................... 271 10.6 Surface Embeddings of Graphs ....................275 Orientable and Nonorientable Surfaces ............. 276 The Euler Characteristic ........................... 278 The Orientable Embedding Conjecture .............. 280 10.7 Related Reading . .................................282 Graph Minors ....................................... 282 Linkages ............................................. 282 Brambles ............................................ 283 Matroids and Duality ............................... 284 Matroid Minors ..................................... 284 10.1 Plane and Planar Graphs A graph is said to be embeddable in the plane,orplanar, if it can be drawn in the plane so that its edges intersect only at their ends. Such a drawing is called 244 10 Planar Graphs a planar embedding of the graph. A planar embedding G of a planar graph G can be regarded as a graph isomorphic to G; the vertex set of G is the set of points representing the vertices of G, the edge set of G is the set of lines representing the edges of G, and a vertex of G is incident with all the edges of G that contain it. For this reason, we often refer to a planar embedding G of a planar graph G as a plane graph, and we refer to its points as vertices and its lines as edges. However, when discussing both a planar graph G and a plane embedding G of G, in order to distinguish the two graphs, we call the vertices of G points and its edges lines; thus, by the point v of G we mean the point of G that represents the vertex v of G, and by the line e of G we mean the line of G that represents the edge e of G. Figure 10.1b depicts a planar embedding of the planar graph K5 \ e, shown in Figure 10.1a. (a) (b) Fig. 10.1. (a) The planar graph K5 \ e, and (b) a planar embedding of K5 \ e The Jordan Curve Theorem It is evident from the above definition that the study of planar graphs necessarily involves the topology of the plane. We do not attempt here to be strictly rigorous in topological matters, however, and are content to adopt a naive point of view toward them. This is done so as not to obscure the combinatorial aspects of the theory, which is our main interest. An elegant and rigorous treatment of the topological aspects can be found in the book by Mohar and Thomassen (2001). The results of topology that are especially relevant in the study of planar graphs are those which deal with simple curves. By a curve, we mean a continuous image of a closed unit line segment. Analogously, a closed curve is a continuous image of a circle. A curve or closed curve is simple if it does not intersect itself (in other words, if the mapping is one-to-one). Properties of such curves come into play in the study of planar graphs because cycles in plane graphs are simple closed curves. A subset of the plane is arcwise-connected if any two of its points can be connected by a curve lying entirely within the subset. The basic result of topology that we need is the Jordan Curve Theorem. 10.1 Plane and Planar Graphs 245 Theorem 10.1 The Jordan Curve Theorem Any simple closed curve C in the plane partitions the rest of the plane into two disjoint arcwise-connected open sets. Although this theorem is intuitively obvious, giving a formal proof of it is quite tricky. The two open sets into which a simple closed curve C partitions the plane are called the interior and the exterior of C. We denote them by int(C) and ext(C), and their closures by Int(C)andExt(C), respectively (thus Int(C) ∩ Ext(C)=C). The Jordan Curve Theorem implies that every arc joining a point of int(C)toa point of ext(C) meets C in at least one point (see Figure 10.2). C int(C) ext(C) Fig. 10.2. The Jordan Curve Theorem Figure 10.1b shows that the graph K5 \e is planar. The graph K5, on the other hand, is not planar. Let us see how the Jordan Curve Theorem can be used to demonstrate this fact. Theorem 10.2 K5 is nonplanar. Proof By contradiction. Let G be a planar embedding of K5, with vertices v1,v2,v3,v4,v5. Because G is complete, any two of its vertices are joined by an edge. Now the cycle C := v1v2v3v1 is a simple closed curve in the plane, and the vertex v4 must lie either in int(C)orinext(C). Without loss of generality, we may suppose that v4 ∈ int(C). Then the edges v1v4,v2v4,v3v4 all lie entirely in int(C), too (apart from their ends v1,v2,v3) (see Figure 10.3). Consider the cycles C1 := v2v3v4v2, C2 := v3v1v4v3,andC3 := v1v2v4v1. Observe that vi ∈ ext(Ci), i =1, 2, 3. Because viv5 ∈ E(G)andG is a plane graph, it follows from the Jordan Curve Theorem that v5 ∈ ext(Ci), i =1, 2, 3, too. Thus v5 ∈ ext(C). But now the edge v4v5 crosses C, again by the Jordan Curve Theorem. This contradicts the planarity of the embedding G. A similar
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