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Graph Theory Ashay Dharwadker Shariefuddin Pirzada ISBN 1466254998 COPYRIGHT © 2011 INSTITUTE OF MATHEMATICS H-501 PALAM VIHAR, GURGAON, HARYANA 122017, INDIA www.dharwadker.org Preface Graph theory, as a branch of mathematics, has a glorious history: from Euler’s seven bridges of Königsberg in 1756, to the elusive proof of the four colour theorem in 2000, and beyond. Graph theory is of practical importance to computer scientists for designing efficient algorithms, because many of the hardest (NP-complete) problems are essentially graph theoretic in nature. In recent times, graph theory has found diverse and often unexpected applications in science, engineering and technology. Seemingly difficult problems become easy to solve when expressed in the proper graph theoretical context. The powerful combinatorial methods of graph theory are being used to discover and prove significant new results in a variety of areas of pure mathematics itself. Indeed, the past few decades have witnessed a rising level of interest and growing activity among mathematicians, scientists and engineers in graph theory. As a result, one finds graph theory as a vital component of the mathematics curriculum in colleges and universities all over the world. In India, the model syllabus for graduate level mathematics proposed by the University Grants Commission includes graph theory as a recommended course. This book has grown from our experience over the past several years in teaching various topics in graph theory, at both the graduate and undergraduate levels. As the number of students opting for graph theory is rapidly increasing, an attempt has been made to provide the latest and best available information on the subject. Our aim is to present the basics of graph theory in such a way that an average student can acquire as much depth and comprehension as possible in a first course. The book is primarily intended for use as a textbook at the graduate level (for students pursuing masters in mathematics and computer science), with the first eleven chapters forming a one year course. However, the first eight chapters may be used as a one semester course at the undergraduate level for students of computer science and engineering. The final sections of many chapters introduce advanced topics and unsolved problems that are the object of current research in graph theory. Thus, the book can also be used by students pursuing research work in M. Phil and Ph. D. programmes. There are many new topics in this book that have not appeared before in print: new proofs of various classical theorems, signed degree sequences, criteria for graphical sequences, eccentric sequences, matching and decomposition of planar graphs into trees. Scores in digraphs appear for the first time in print and the climax of the book is a new proof of the famous four colour theorem. Many earlier books, monographs and articles have been used in the preparation of this book and we have included a comprehensive bibliography at the end of the book. We would like to thank the Canadian Mathematical Society and the Math Forum at Drexel University for announcing the new proof of the four colour theorem in 2000. We are extremely grateful to the University of Kashmir and the Institute of Mathematics (Gurgaon) for their support during the writing of this book. Our sincere thanks go to Merajuddin (AMU, Aligarh), M. A. Sofi (University of Kashmir), Petrovic Vojislav (Novi Sad University), Ivanyi Antal (Eotvos Lorand University, Hungary), Zhou Guofei (Nanjing University, China), M. R. Sridharan (IIT, Kanpur), V. Krishnamurthy (BITS, Pilani), Niels Karlsson (Akureyri University), John-Tagore Tevet and Jüri Martin (Eurouniversity, Tallinn), Anita Pasotti (Universita degli Studi di Brescia) and Vladimir Khachatryan (SUNY, Stony Brook) for their valuable suggestions and encouragement. We thank all our friends and colleagues, especially T. A. Chishti (University of Kashmir) and all the members of the Institute of Mathematics (Gurgaon) for supporting our work. We thank our research students, especially T.A. Naikoo for help in drawing the figures and carefully proof reading the manuscript. We are grateful to our families for their love and support during the time this book was being written. Finally, it is a pleasure to thank the management and staff of Orient Longman and Universities Press (India) for their interest, cooperation, and fine workmanship. Ashay Dharwadker Shariefuddin Pirzada Contents 1. Introduction 1 2. Degree Sequences 37 3. Eulerian and Hamiltonian Graphs 59 4. Trees 79 5. Connectivity 105 6. Planarity 134 7. Colourings 163 8. Matchings and Factors 198 9. Edge Graphs and Eccentricity Sequences 234 10. Graph Matrices 263 11. Digraphs 296 12. The Four Colour Theorem 335 13. Graph Algorithms 366 14. Score Structure in Digraphs 403 Bibliography 442 1. Introduction Graph theory owes its evolution to the study of some physical problems involving sets of objects and binary relations among them. It is difficult to pinpointits formulationto a singlesource; infact, graph theorycan be saidtohave been discovered many times,each discovery beingindependentof theother. The earliest knownstudiesappear inthe works ofEuler,Kirchoff,CayleyandHamilton.Thetwentiethcenturywitnessedconsiderableac- tivityinthisarea, withnewdiscoveriesandproofsbeingproposedassolutionstoclassical problemsincludingthecelebratedfourcolourproblem. 1.1 Basic Concepts LetV be a nonemptyset. The cartesian product ofV with itself, denoted byV×V, is the setof all unordered pairs of elementsofV. That is,V×V ={(u, v):u, v∈V}. We denote byV(2) thesetofunorderedpairsofdistinctelementsofV,byV[2]thesetofunorderedpairs of elements ofV, not necessarily distinct, and byV(2) the set of ordered pairs of distinct elementsofV. Definition: Asimplegraph(orbriefly,agraph)GisafinitenonemptysetV togetherwith asymmetric,irreflexiverelationRonV.TheelementsofthesetV arecalledtheverticesof thegraph andthe relationR iscalled the adjacency relation.If uisrelated to v byR, then u is said to be adjacent to v and we write uRv. An example of a simple graph is given in Figure1.1(a). Since R isa symmetricrelation, itdefines a subsetE ofV(2). The elements of the set E arecalledtheedgesofthegraph. (a)Simplegraph (b)Multigraph (c)Infinitegraph Fig.1.1 2 Introduction From the above definition, we observe that a graph G is a pair (V, E), where V is a nonemptyset whose elements are called the vertices of G and E is a subset ofV(2) whose elementsarecalledtheedgesofG. We observe that there is an incidence relation I between the vertex setV and the edge set E of a graph. If the element e∈E, there is a pair of distinctvertices u and v such that e={u, v}. The vertices u and v are called end vertices of e, and u and v are said to be incidentwithe(uIeandvIe).Also,eissaidtobeincidentwithuandv,andinthatcasewe writeeI0uandeI0v,whereI0 istherelationconversetoI. A graph with a finite number of vertices and finite number of edges is called a finite graph,otherwiseitisaninfinitegraph(Figure1(c)). We representagraphGwithvertex setV andedgesetE by(V(G), E(G)). Since weare onlygoingtodealwithfinitegraphs,wewriteV(G)={v1,v2,...,vn},E(G)={e1,e2,...,em}. We define |V|=nto be the order of G and |E|=m tobe the sizeof G. Such a graph is called an (n, m) graph. If there is an edge e between the vertices u and v, we briefly write e=uv and say edge e joins the vertices u and v. A vertex is said to be isolated if it is not adjacenttoanyothervertex. Multigraph: Amultigraphisapair(V,E),whereV isanonemptysetofverticesandE is amultisetofedges,beingamulti-subsetofV[2].Thenumberoftimesanedgee=uvoccurs in E is called the multiplicityof e and edges withmultiplicitygreater than one are called multipleedges.AnexampleofamultigraphisgiveninFigure1(b). Generalgraph: A general graphisapair (V,E), whereV isa nonemptysetof Moment andE isamultisetofedges,beingamulti-subsetofV(3).Anedgeoftheforme=uu, u∈V iscalledaloop.Anedgewhichisnotaloopiscalledaproperedgeorlink.Thenumberof timesedgeeoccursiscalleditsmultiplicity,andproperedgeswithmultiplicitygreaterthan onearecalledmultipleedges.Loopswithmultiplicitygreater thanonearecalledmultiple loops(Fig.1.2). If u, v∈V in a general graph or multigraph G, then the multiplicity of the edge uv is denoted by q [u, v]. If uv is not an edge, then q [u, v]=0. Similarly, if u is a vertex of a G G generalgraphG,thenthenumberofloopsatuinGisdenotedbyq [u]. G ThegraphobtainedbyreplacingallmultipleedgesbysingleedgesinamultigraphGis calledthe underlyinggraphof G. Similarly,if G isa general graph, thegraph H obtained by removing all its loops and replacing all multiple edges by single edges is called the underlyinggraphofG. Fig.1.2 Generalgraph GraphTheory 3 1.2 Degrees Ina graphG, thedegree of a vertex v isthe numberof edges of G whichare incidenttov andisdenotedbyd(v)ord(v|G).We haved(v)=|{e∈E:e=uv,foru∈V}|.Theminimum degree and the maximum degree of a graph G are denoted byδ(G)and ∆(G) respectively. InthegraphofFigure1.3(a),d(v1)=d(v3)=d(v6)=3, d(v2)=1, d(v4)=0andd(v5)=2. A graph is said to be regular if all its vertices are of same degree and k-regular if all itsvertices are of degree k. A 3-regular graph is also called a cubic graph. A vertex with degree zero is an isolated vertex, a vertex with degree one is a pendant vertex and the uniqueedge incident to a pendant vertex is a pendant edge. A vertex of odd degree is an oddvertex andavertex ofevendegree isan evenvertex. InFigure1.3(a), v4 isanisolated vertex and v2 is a pendant vertex. The graph shown in Figure 1.3(b) is 2-regular and that giveninFigure1.3(c)is3-regular, i.e., cubic. In a general graph G, a loop incident to a vertex v is counted as two edges incident to v. Therefore d(v) is the number of non-loop edges incident to v plus twice the number of loopsatv. (a) (b) (c) Fig.1.3 Graphs(b)and(c)areregular ThefollowingresultisduetoEuler[76]. Theorem 1.1 The sum of the degrees of a graph is even, being twice the number of edges. Proof Letmbe thenumberofedgesina graphG=(V, E). Since each edgecontributes twoto the degrees, one at the beginning vertex and one at the end vertex of the edge, the sumofthedegreesisevenandequaltotwicethenumberofedges.Hence, ∑d(v)=2m. q vεV Theorem1.2 Inanygraphthereisanevennumberofverticesofodddegree. Proof LetG=(V, E)beagraphandd(v)bethedegreeofthevertexv∈V.Let|E|=m. Then ∑ d(v)=2mandtherefore, vεV 4 Introduction ∑ d(v ) + ∑ d(v )=2m. j k odddegree evendegree (1.2.1) vertices vertices I II Since therighthandsideof (1.2.1) iseven, and(II) in(1.2.1) isalsoeven, therefore (I) in(1.2.1) iseven. Hence, ∑d(v )=even. j odddegree vertices Thisisonlypossiblewhenthenumberofverticeswithodddegreeiseven. q 1.3 Isomorphism Let G and H be general graphs. Let f be a one−one mapping of V(G) onto V(H), and g be a one−one mapping of E(G) onto E(H). Let θ denote the ordered pair (f, g). Then θ is an isomorphismof G onto H, when the vertex x is incident withthe edge e in G if and only if the vertex fx is incident with the edge ge in H (Fig. 1.4). If such an isomorphism θexists,thegraphsGandH aresaidtobeisomorphicandisdenotedbyG∼=H. Wehave, |V(G)|=|V(H)|and|E(G)|=|E(H)|. e ge x fx G H Fig.1.4 We can think θ to be an operation transforming G into H and write θG= H. Also, we writeθv= fv and θe=ge for each vertex v and each edge e of G. Clearly, G and H can berepresentedbythesamediagram.TherepresentativeofanedgeorvertexxofGcanbe reinterpretedastherepresentativeofθxinH. Anisomorphismof agraph Gontoitselfiscalledan automorphismofG. AnygraphG hasthe identicalor trivialautomorphismI suchthat Ix=x for each edge or each vertex x ofG. Clearly, two graphs G and G0 are isomorphic to each other if there is a one−one cor- respondence between theirvertices, and between theiredges such thatthe incidence rela- tionshipis preserved. For example, the graphs shown in Figure 1.5(a), Figure 1.5(b) and Figure1.5(c)are isomorphic. GraphTheory 5 (a)Isomorphicgraphs (b)Isomorphicgraphs (c)Isomorphicgraphs Fig.1.5 Itfollowsfromthedefinitionofisomorphismthattwoisomorphicgraphshave i. thesamenumberofvertices, ii. thesamenumberofedges,and iii. anequalnumberofverticeswithagivendegree. However, theseconditionsare notsufficient.To seethis,considerthetwographsgivenin Figure1.6. Fig.1.6 Non-isomorphicgraphs These graphs satisfy all the three conditions, but they are not isomorphic because the vertex x in(a) correspondstovertex yin(b) asthere are noothervertices ofdegree three, and in (b) there is only one pendant vertex w adjacent to y, while in (a) there are two pendantverticesuandvadjacenttox. ThegraphsinFigure.1.7alsosatisfyconditions(i),(ii)and(iii)butarenotisomorphic. 6 Introduction Fig.1.7 Non-isomorphicgraphs Wehave thefollowingobservationonisomorphismofgraphs. Theorem1.3 Therelationisomorphismingraphsisanequivalencerelation. Proof The relation of isomorphism between graphs is reflexive because of the trivial automorphisms. Letθ=(f, g)beanisomorphismofagraphGontoagraphH,soG∼=H. Thenthereis aninverseisomorphismθ−1=(f−1,g−1)ofHontoG.SoH∼=G.Therefore∼=issymmetric. Now,letθ=(f,g)beanisomorphismofGontoH,andφθ=(f1f,g1g)ofGontoK.Here f1f isamappingobtainedbyapplyingfirst f andthen f1.Similarly,φθistheisomorphism obtainedapplyingfirstθandthenφ.Thus∼=istransitive. Hencetherelationisomorphismisanequivalencerelation. q Remark Themultiplicationoftheisomorphismdefinedaboveisassociative. Since the relation isomorphism is an equivalence relation, it partitions the class of all graphsintodisjointnonemptysubclassescalledisomorphismclasses,suchthattwographs belongtothesameisomorphismclassifandonlyiftheyareisomorphic. Theorem1.4 Let G and H be graphs and let f be a one−one mappingV(G) ontoV(H) such that two distinct vertices x and y of G are adjacent if and only if the corresponding vertices fx and fy of H are adjacent in H. Then there is a uniquely determined one−one mappinggofE(G)ontoE(H)suchthat(f,g)isanisomorphismofGontoH. Proof Let e be any edge of G, having distinct ends x and y. By hypothesis, there is a uniquelydeterminededgee0 ofHwhoseendsare fxand fy.Wedefineaone−onemapping gbytherulege=e0,foreachedgeeofG.Itisthenclearthat(f,g)isanisomorphismofG ontoH. Conversely,letgbeamappingsuchthat(f,g)isanisomorphismofGontoH.Thenfor eachedgeeofG,thereisanedgee0 ofH suchthatge=e0. q AnisomorphismofagraphGontoagraphH isdefinedasaone−onemappingofV(G) ontoV(H) that preserves adjacency. This specialisationcan be regarded as an application ofTheorem1.4. Definition: Two graphs G(V, E) and H(U, F) are label-isomorphicif and onlyifV =U, and for any pair u, v in V, uv ∈E if and only if uv∈ F. The graphs of Figure 1.8 are isomorphic,butnotlabel-isomorphic.