PrefacexiAboutthethird editionThere is no denying that this book has grown. Is it still as lean andconcentrating on the essential' as I said it should be when I wrote thepreface to the first edition, now almost eight years ago?Ibelieve that it is, perhaps now more than ever. So why the increasein volume? Part of the answer is that I have continued to pursue theoriginal dual aim of offering two different things between one pairofcovers:areliablefrst introduction tographtheory thatcan beused eitherfor personal study or as a course text;duate text that also offers some depth on thamaimportantopics.For each of these aims, some material has been added. Some of thiscoversnetopics,which can be included or skipped as desired.Anexample at the introductory level is the new section on packing andcovering with the Erdos-Posa theorem, or the inclusion of the stable in the matching chapter. An example at the graduategethlevel is the Robertson-Seymour structure theorem for graphs without agiven minor: a result that takes a few lines to state, but one which is in-creasingly relied on in the literature, so that an easily accessible referenceseems desirable. Another addition, also in the chapter on graph mis a new proof of the Kuratowski theorem for higher surfaces-aproofwhich illustrates the interplay between graph minor theory and surfacetopology better than was previously possible. The proof is complementedby an appendix on surfaces, which supplies the required background andnore light on the proof of the graph minor theoremalsoshesomeChanges that affect previously existing material are rare, except forcountless local improvements intended to consolidate and polish ratherthan change. I am aware that, as this book is increasingly adopted asa course text, there is a certain desire for stability. Many of these localimprovementsare the result of generousfeedbackIgot from colleagueusing thebook in this way,andIam verygratefulforthirhelp andadviceThere are also some local additions.Most of these developed frommy own notes, pencilled in the margin as I prepared to teach from thebook.They typically complement an important but technical proof,when I felt that its essential ideas might get overlooked in the formalwrite-up.For example,theproof of theErdos-Stonetheorem nowhasininformal post-mortemthatlooksathowexactlytheregularitylemmcomes to be applied in it. Unlike the formal proof, the discussion startsoutfromthemain idea,andfinallyarrives athowtheparametersto bedeclared at the start of the formal proof must be specified. Similarlythere is now a discussion pointing to some ideas in the proof of the perfec
Preface xi About the third edition There is no denying that this book has grown. Is it still as ‘lean and concentrating on the essential’ as I said it should be when I wrote the preface to the first edition, now almost eight years ago? I believe that it is, perhaps now more than ever. So why the increase in volume? Part of the answer is that I have continued to pursue the original dual aim of offering two different things between one pair of covers: • a reliable first introduction to graph theory that can be used either for personal study or as a course text; • a graduate text that also offers some depth on the most important topics. For each of these aims, some material has been added. Some of this covers new topics, which can be included or skipped as desired. An example at the introductory level is the new section on packing and covering with the Erd˝os-P´osa theorem, or the inclusion of the stable marriage theorem in the matching chapter. An example at the graduate level is the Robertson-Seymour structure theorem for graphs without a given minor: a result that takes a few lines to state, but one which is increasingly relied on in the literature, so that an easily accessible reference seems desirable. Another addition, also in the chapter on graph minors, is a new proof of the ‘Kuratowski theorem for higher surfaces’—a proof which illustrates the interplay between graph minor theory and surface topology better than was previously possible. The proof is complemented by an appendix on surfaces, which supplies the required background and also sheds some more light on the proof of the graph minor theorem. Changes that affect previously existing material are rare, except for countless local improvements intended to consolidate and polish rather than change. I am aware that, as this book is increasingly adopted as a course text, there is a certain desire for stability. Many of these local improvements are the result of generous feedback I got from colleagues using the book in this way, and I am very grateful for their help and advice. There are also some local additions. Most of these developed from my own notes, pencilled in the margin as I prepared to teach from the book. They typically complement an important but technical proof, when I felt that its essential ideas might get overlooked in the formal write-up. For example, the proof of the Erd˝os-Stone theorem now has an informal post-mortem that looks at how exactly the regularity lemma comes to be applied in it. Unlike the formal proof, the discussion starts out from the main idea, and finally arrives at how the parameters to be declared at the start of the formal proof must be specified. Similarly, there is now a discussion pointing to some ideas in the proof of the perfect
xiiPrefacegraph theorem. However, in all these cases the formal proofs have beenleft essentially untouched.The only substantial change to existing material is that the oldTheorem 8.1.1 (that cr?n edges force a TKr) seems to have lost itsnice (and long) proof. Previously, this proof had served as a welcomeortunitytoexplain methods in sparse extremal graph theory.someThese methods have migrated to the connectivity chapter, where theynow live under theroofof the new proof by Thomas and Wollan that 8knedges make a 2k-connected graphlinked.Sothey arestll there, leanerthan ever before, and just presenting themselves under a new guise. Asa consequence of this change, the two earlier chapters on dense andsparseextremalgraphtheorycouldbereunitedoformanewchapterappropriately named as Eartremal Graph TheoryFinally, there is an entirely new chapter, on infinite graphs. Whengraph theory first emerged as a mathematical discipline, finite and infinite graphsere usually treated on a par. This has changed in recentyears, whichIseeasarerettableloss: infinitegraphs continuetopvide a natural and frequently used bridge to other fields of mathematicsand they hold some special fascination of their own. One aspect of thisis that proofs often have to be more constructive and algorithmic innature than their finite counterparts. The infinite version of Menger'stheorem in Section 8.4 is a typical example: it offers algorithmic insightsinto connectivityproblems networks that are invisible to the slick1inductiveproofsofthefinitetheoremgiveninChapter3.3Oncee more, my thanks go to all the readers and colleagues whosemments helped toimprove the book.Iam particularly grateful to ImreLeader for his judicious comments on the whole of the infinite chapter; tomygraphtheoryseminarnparticularoLilianMatthiesenandPhilippSprissel, for giving the chapter a test run and solving all its exercises(of which eighty survived their scrutiny); to Agelos Georgakopoulos formuch proofreading elsewhere; to Melanie Win Myint for recompiling theindex and extending it substantially:and to Tim Stelldinger for nursingthe whale on page 404 until it was strong enough to carry its babydinosaur.May 2005RD
xii Preface graph theorem. However, in all these cases the formal proofs have been left essentially untouched. The only substantial change to existing material is that the old Theorem 8.1.1 (that cr2n edges force a TKr) seems to have lost its nice (and long) proof. Previously, this proof had served as a welcome opportunity to explain some methods in sparse extremal graph theory. These methods have migrated to the connectivity chapter, where they now live under the roof of the new proof by Thomas and Wollan that 8kn edges make a 2k-connected graph k-linked. So they are still there, leaner than ever before, and just presenting themselves under a new guise. As a consequence of this change, the two earlier chapters on dense and sparse extremal graph theory could be reunited, to form a new chapter appropriately named as Extremal Graph Theory. Finally, there is an entirely new chapter, on infinite graphs. When graph theory first emerged as a mathematical discipline, finite and infi- nite graphs were usually treated on a par. This has changed in recent years, which I see as a regrettable loss: infinite graphs continue to provide a natural and frequently used bridge to other fields of mathematics, and they hold some special fascination of their own. One aspect of this is that proofs often have to be more constructive and algorithmic in nature than their finite counterparts. The infinite version of Menger’s theorem in Section 8.4 is a typical example: it offers algorithmic insights into connectivity problems in networks that are invisible to the slick inductive proofs of the finite theorem given in Chapter 3.3. Once more, my thanks go to all the readers and colleagues whose comments helped to improve the book. I am particularly grateful to Imre Leader for his judicious comments on the whole of the infinite chapter; to my graph theory seminar, in particular to Lilian Matthiesen and Philipp Spr¨ussel, for giving the chapter a test run and solving all its exercises (of which eighty survived their scrutiny); to Agelos Georgakopoulos for much proofreading elsewhere; to Melanie Win Myint for recompiling the index and extending it substantially; and to Tim Stelldinger for nursing the whale on page 404 until it was strong enough to carry its baby dinosaur. May 2005 RD
xiiPrefaceAboutthefourtheditionIn this fourth edition there are few substantial additions of new material, but many improvements.As with previous new editions, there are countless small and subtlechanges to further elucidate a particular argument or concept. Whenprompted by reader feedback, for which I am always grateful, I still tryto recast details that have been found harder than they should be. Thesecan be very basic; a nice example, this time, is the definition of a minorin ChapterAtamoresubostantiallevel.thereralnewandsimareseyDlerproof classical results, in one case reducing the already shortened earlierproof to halfits length (and twice its beauty),These newly added proofsinclude the marriage theorem, the tree packing theorem, Tutte's cyclespace and wheel theorem, Fleischner's theorem on Hamilton cycles, andthe threshold theorem for the edge probability guaranteeing a specifiedtype of subgraph.Thereare also one or two genuinely new theoremsOne of theseious local degree cconditionfortheexistenceofaisaninHamilton cycle, due to Asratian and Khachatrian, that implies a numberof classicalnicity theoremsIn some sections I have reorganized the material slightly, or rewrten the narrative. Typically, these are sections that had grown over theprevious three editions, and this was beginning to affect their balanceof material and momentum. As the book remains committed to offeringnot just a collection of theorems and proofs, but tries whenever possibleto indicate a somewhat larger picture in which these have their placemaintaining its original freshness and flow remains a challenge that Ienjoy trying to meet.Finally, the book has its own dedicated website now, athttp://diestel-graph-theory.com/Potentially, this offers opportunities for more features surrounding thebookthan thetraditional free online edition and adwindling collectionofmisprints.Ifyouhaveanyideasand wouldliketoseethemimplementeddo let me know.RDMay 2010
Preface xiii About the fourth edition In this fourth edition there are few substantial additions of new material, but many improvements. As with previous new editions, there are countless small and subtle changes to further elucidate a particular argument or concept. When prompted by reader feedback, for which I am always grateful, I still try to recast details that have been found harder than they should be. These can be very basic; a nice example, this time, is the definition of a minor in Chapter 1. At a more substantial level, there are several new and simpler proofs of classical results, in one case reducing the already shortened earlier proof to half its length (and twice its beauty). These newly added proofs include the marriage theorem, the tree packing theorem, Tutte’s cycle space and wheel theorem, Fleischner’s theorem on Hamilton cycles, and the threshold theorem for the edge probability guaranteeing a specified type of subgraph. There are also one or two genuinely new theorems. One of these is an ingenious local degree condition for the existence of a Hamilton cycle, due to Asratian and Khachatrian, that implies a number of classical hamiltonicity theorems. In some sections I have reorganized the material slightly, or rewritten the narrative. Typically, these are sections that had grown over the previous three editions, and this was beginning to affect their balance of material and momentum. As the book remains committed to offering not just a collection of theorems and proofs, but tries whenever possible to indicate a somewhat larger picture in which these have their place, maintaining its original freshness and flow remains a challenge that I enjoy trying to meet. Finally, the book has its own dedicated website now, at http://diestel-graph-theory.com/ Potentially, this offers opportunities for more features surrounding the book than the traditional free online edition and a dwindling collection of misprints. If you have any ideas and would like to see them implemented, do let me know. May 2010 RD
xivPrefaceAboutthefiftheditionThis ffth edition of thee book is again a major overhaul, in the spirit ofits first and third edition.I have rewritten Chapter 12 on graph minors to take account ofrecent developments.In addition to many smaller updates it offersnew proof ofthetree-width duality theorem.duetoMazoit.whichhas not otherwise been published. More fundamentally.I have addeda sectionDntangles.OriginallydevisedbyRobertsonandSeymourasal device for their proof of the graphrti1,tangleshave turned out to beamental than this: they defirnoa new paradigm for identifying highly connected parts in a graph. Un-like earlier attempts at defining such substructuresin terms of, sayhighly connected subgraphs, minors, or topological minotanglesdonot attempt to pin down this substructure in terms of vertices, edges, orconnecting paths,but seek to captureit indirectly by orienting all thelow-order separations of the graph towards itIn short.we no longer askwhat exactly the highly connected region is.but only where it is.Fors, this is exactly what matters. Moreoover.thismabstonofhighlocalconnectivity can eorted toSVDecontexts outside graph theory. This, in turn, makes graph minoseorapplicable beyond graph theory itself in a new way, via tangles. I havewritten the new section on tangles from this modern perspective.Chapter 2hasanewlywritten section on tree packand coveringI rewrote it from scratch to take advantage of a beautiful new unifiedtheorem containbothaspectsatonce:thepacking-coveringtheoremof BowlerandCarmesin.Whiletheiroriginal result wasproved formtroids, its graph version hasavery shortand self-contained proof.Thisn in Chapter 2.4, and again is not fproof is givend in print elsewheChapter 8, on infinitegraphs, now treats the topological aspects oflocally finite graphs more thoroughly. It puts the Freudenthal compactification of a graph G into perspective by describing it, in addition, asinverse limit of the finite contractionnors of G.Readers withbackground in group theory will find this familiar.Asalways,thereare countless small improvementstothenarrativeand exercises. My thanksgo to all those who suggested thesDroofsFinally, I have made two adjustments to help ensure that the ex.sahleinataimeofinstaTheHints appendix still exists, but has been relegated to the professionalelectronic edition so that lecturers can decide which hints to give andwhich not. Similarly, exercises asking for a proof of a named theorem nolonger mention this name, so that the proof cannot simply be searchedfor. However if you know the name and wish to find the exercise, theindex still has a name entry that will take you to the right pageRDJuly 2016
xiv Preface About the fifth edition This fifth edition of the book is again a major overhaul, in the spirit of its first and third edition. I have rewritten Chapter 12 on graph minors to take account of recent developments. In addition to many smaller updates it offers a new proof of the tree-width duality theorem, due to Mazoit, which has not otherwise been published. More fundamentally, I have added a section on tangles. Originally devised by Robertson and Seymour as a technical device for their proof of the graph minor theorem, tangles have turned out to be much more fundamental than this: they define a new paradigm for identifying highly connected parts in a graph. Unlike earlier attempts at defining such substructures—in terms of, say, highly connected subgraphs, minors, or topological minors—tangles do not attempt to pin down this substructure in terms of vertices, edges, or connecting paths, but seek to capture it indirectly by orienting all the low-order separations of the graph towards it. In short, we no longer ask what exactly the highly connected region is, but only where it is. For many applications, this is exactly what matters. Moreover, this more abstract notion of high local connectivity can easily be transported to contexts outside graph theory. This, in turn, makes graph minor theory applicable beyond graph theory itself in a new way, via tangles. I have written the new section on tangles from this modern perspective. Chapter 2 has a newly written section on tree packing and covering. I rewrote it from scratch to take advantage of a beautiful new unified theorem containing both aspects at once: the packing-covering theorem of Bowler and Carmesin. While their original result was proved for matroids, its graph version has a very short and self-contained proof. This proof is given in Chapter 2.4, and again is not found in print elsewhere. Chapter 8, on infinite graphs, now treats the topological aspects of locally finite graphs more thoroughly. It puts the Freudenthal compactification of a graph G into perspective by describing it, in addition, as an inverse limit of the finite contraction minors of G. Readers with a background in group theory will find this familiar. As always, there are countless small improvements to the narrative, proofs, and exercises. My thanks go to all those who suggested these. Finally, I have made two adjustments to help ensure that the exercises remain usable in class at a time of instant internet access. The Hints appendix still exists, but has been relegated to the professional electronic edition so that lecturers can decide which hints to give and which not. Similarly, exercises asking for a proof of a named theorem no longer mention this name, so that the proof cannot simply be searched for. However if you know the name and wish to find the exercise, the index still has a name entry that will take you to the right page. July 2016 RD
ContentsDrefaceVI1. The Basics1.1 Graphs*1.2 The degree ofL1.3 Paths and cycles*1.4Connectivity10131.5 TreesO441.6 Bipartite graphs*117-1.8 Euler tours*a231.9SorJinearalo1.10 Other27notionsofgraphsExercises30Note2. Matching, Covering and Packing352.1 Matching in bipartite graphs*362.2 Matching in general graphs(412.3 The Erd&s-P6sa theorem452.4 Tree packing and arboricity522.5 Path covers+++++++++++Exercises53SectionsmaarerecommendedfoOf.ionemarthelxinninmencedfohrst
Contents Preface ................................................................ vii 1. The Basics ...................................................... 1 1.1 Graphs* ........................................................ 2 1.2 The degree of a vertex* ......................................... 5 1.3 Paths and cycles* .............................................. 6 1.4 Connectivity* .................................................. 10 1.5 Trees and forests* .............................................. 13 1.6 Bipartite graphs* ............................................... 17 1.7 Contraction and minors* ....................................... 19 1.8 Euler tours* .................................................... 22 1.9 Some linear algebra ............................................ 23 1.10 Other notions of graphs ........................................ 27 Exercises ....................................................... 30 Notes .......................................................... 33 2. Matching, Covering and Packing ............................. 35 2.1 Matching in bipartite graphs* .................................. 36 2.2 Matching in general graphs(∗) ................................... 41 2.3 The Erd˝os-P´osa theorem ....................................... 45 2.4 Tree packing and arboricity ..................................... 48 2.5 Path covers .................................................... 52 Exercises ....................................................... 53 Notes .......................................................... 56 * Sections marked by an asterisk are recommended for a first course. Of sections marked (∗), the beginning is recommended for a first course