Lecture Notes on GRAPH THEORY Tero Harju Department of Mathematics University of Turku FIN-20014 Turku, Finland e-mail: harju@utu.fi 2007 Contents 1 Introduction 2 1.1 Graphs and their plane figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.2 Subgraphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.3 Paths and cycles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2 Connectivity of Graphs 16 2.1 Bipartite graphs and trees. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 2.2 Connectivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 3 Tours and Matchings 30 3.1 Eulerian graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 3.2 Hamiltonian graphs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 3.3 Matchings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 4 Colourings 43 4.1 Edge colourings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 4.2 Ramsey Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 4.3 Vertex colourings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 5 Graphs on Surfaces 60 5.1 Planar graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 5.2 Colouring planar graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 5.3 Genus of a graph . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 6 Directed Graphs 83 6.1 Digraphs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 6.2 Network Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 Index 96 1 Introduction Graph theory can be said to have its beginning in 1736 when EULER considered the (general case of the) Königsberg bridge problem: Is there a walk- ing route that crosses each of the seven bridges of Königsberg exactly once? (Solutio Problema- tis ad geometriam situs pertinentis, Commentarii Academiae Scientiarum Imperialis Petropolitanae 8 (1736), pp. 128-140.) It took 200 years before the first book on graph theory was written. This was done by KÖNIG in 1936. (“Theorie der endlichen und unendlichen Graphen”, Teubner, Leipzig, 1936. Translation in English, 1990.) Since then graph theory has developed into an extensive and popular branch of mathematics, which has been applied to many problems in mathematics, computer science, and other scientific and not-so-scientific areas. For the history of early graph theory, see N.L. BIGGS, R.J. LLOYD AND R.J. WILSON, “Graph Theory 1736 – 1936”, Clarendon Press, 1986. There seem to be no standard notations or even definitions for graph theoretical objects. This is natural, because the names one uses for these objects reflect the applications. So, for instance, if we consider a communications network (say, for email) as a graph, then the computers, which take part in this network, are called nodes rather than vertices or points. On the other hand, other names are used for molecular structures in chemistry, flow charts in programming, human relations in social sciences, and so on. These lectures study finite graphs and majority of the topics is included in J.A. BONDY AND U.S.R. MURTY, “Graph Theory with Applications”, Macmillan, 1978. R. DIESTEL, “Graph Theory”, Springer-Verlag, 1997. F. HARARY, “Graph Theory”, Addison-Wesley, 1969. D.B. WEST, “Introduction to Graph Theory”, Prentice Hall, 1996. R.J. WILSON, “Introduction to Graph Theory”, Longman, (3rd ed.) 1985. In these lectures we study combinatorial aspects of graphs. For more algebraic topics and methods, see N. BIGGS, “Algebraic Graph Theory”, Cambridge University Press, (2nd ed.) 1993. and for computational aspects, see S. EVEN, “Graph Algorithms”, Computer Science Press, 1979. 3 In these lecture notes we mention several open problems that have gained respect among the researchers. Indeed, graph theory has the advantage that it contains easily formulated open problems that can be stated early in the theory. Finding a solution to any one of these problems is on another layer of difficulty. Sections with a star ( ) in their heading are optional. Notations and notions For a finite set , denotes its size (cardinality, the number of its elements). Let and in general, for integers . For a real number , the floor and the ceiling of are the integers and A family of subsets of a set is a partition of , if and for all different and For two sets and , is their Cartesian product. For two sets and , is their symmetric difference. Here . Two numbers (often and for sets and ) have the same parity, if both are even, or both are odd, that is, if . Otherwise, they have opposite parity. Graph theory has abundant examples of NP-complete problems. Intuitively, a problem is in P 1 if there is an efficient (practical) algorithm to find a solution to it. On the other hand, a problem is in NP 2 , if it is first efficient to guess a solution and then efficient to check that this solution is correct. It is conjectured (and not known) that P NP. This is one of the great problems in modern mathematics and theoretical computer science. If the guessing in NP-problems can be replaced by an efficient systematic search for a solution, then P NP. For any one NP-complete problem, if it is in P, then necessarily P NP. Solvable – by an algorithm – in polynomially many steps on the size of the problem instances. Solvable nondeterministically in polynomially many steps on the size of the problem instances. 1.1 Graphs and their plane figures 4 1.1 Graphs and their plane figures Let be a finite set, and denote by the subsets of of two distinct elements. DEFINITION. A pair with is called a graph (on ). The elements of are the vertices, and those of the edges of the graph. The vertex set of a graph is denoted by and its edge set by . Therefore . In literature, graphs are also called simple graphs; vertices are called nodes or points; edges are called lines or links. The list of alternatives is long (but still finite). A pair is usually written simply as . Notice that then . In order to simplify notations, we also write instead of . DEFINITION. For a graph , we denote and The number of the vertices is called the order of , and is the size of . For an edge , the vertices and are its ends. Vertices and are adjacent or neighbours, if . Two edges and having a common end, are adjacent with each other. A graph can be represented as a plane figure by drawing a line (or a curve) between the points and (representing vertices) if is an edge of . The figure on the right is a drawing of the graph with and . Often we shall omit the identities (names ) of the vertices in our figures, in which case the vertices are drawn as anonymous circles. Graphs can be generalized by allowing loops and parallel (or multiple) edges between vertices to obtain a multigraph , where is a set (of symbols), and is a function that attaches an unordered pair of vertices to each : . Note that we can have . This is drawn in the figure of by placing two (parallel) edges that connect the common ends. On the right there is (a drawing of) a multi- graph with vertices and edges , , , and . 1.1 Graphs and their plane figures 5 Later we concentrate on (simple) graphs. DEFINITION. We also study directed graphs or digraphs , where the edges have a direction, that is, the edges are ordered: . In this case, . The directed graphs have representations, where the edges are drawn as arrows. A digraph can contain edges and of opposite directions. Graphs and digraphs can also be coloured, labelled, and weighted: DEFINITION. A function is a vertex colouring of by a set of colours. A function is an edge colouring of . Usually, for some . If (often ), then is a weight function or a distance function. Isomorphism of graphs DEFINITION. Two graphs and are isomorphic, denoted by , if there exists a bijection such that for all . Hence and are isomorphic if the vertices of are renamings of those of . Two isomorphic graphs enjoy the same graph theoretical properties, and they are often identified. In particular, all isomorphic graphs have the same plane figures (excepting the identities of the vertices). This shows in the figures, where we tend to replace the vertices by small circles, and talk of ‘the graph’ although there are, in fact, infinitely many of such graphs. Example 1.1. The following graphs are iso- morphic. Indeed, the required isomorphism is given by , , , , . Isomorphism Problem. Does there exist an efficient algorithm to check whether any two given graphs are isomorphic or not? The following table lists the number of graphs on a given set of vertices, and the number of nonisomorphic graphs on vertices. It tells that at least for computational purposes an efficient algorithm for checking whether two graphs are isomorphic or not would be greatly appreciated. 1.1 Graphs and their plane figures 6 1 2 3 4 5 6 7 8 9 graphs nonisomorphic Other representations Plane figures catch graphs for our eyes, but if a problem on graphs is to be programmed, then these figures are (to say the least) unsuitable. Matrices of integers are ideal for computers, since every respectable programming language has array structures for these, and computers are good in crunching numbers. Let be ordered. The adjacency matrix of is the -matrix with entries or according to whether or not. For instance, the graphs of Example 1.1 has an adjacency matrix on the right. Notice that the adjacency matrix is always symmetric (with respect to its diagonal consisting of zeros). A graph has usually many different adjacency matrices, one for each ordering of its set of vertices. The following result is obvious from the definitions. Theorem 1.1. Two graphs and are isomorphic if and only if they have a common ad- jacency matrix. Moreover, two isomorphic graphs have exactly the same set of adjacency matrices. Graphs can also be represented by sets. For this, let be a fam- ily of subsets of a set , and define the intersection graph as the graph with vertices , and edges for all and ( ) with . Theorem 1.2. Every graph is an intersection graph of some family of subsets. Proof. Let be a graph, and define, for all , a set Then if and only if . Let be the smallest size of a base set such that can be represented as an inter- section graph of a family of subsets of , that is, for some How small can be compared to the order (or the size ) of the graph? It was shown by KOU, STOCKMEYER AND WONG (1976) that it is algorithmically difficult to determine the number – the problem is NP-complete. 1.2 Subgraphs 7 Example 1.2. As yet another example, let be a finite set of natural numbers, and let be the graph defined on such that if and only if and (for ) have a common divisor . As an exercise, we state: All graphs can be represented in the form for some set of natural numbers. 1.2 Subgraphs Ideally, in a problem the local properties of a graph determine a solution. In such a situation we deal with (small) parts of the graph (subgraphs), and a solution can be found to the problem by combining the information determined by the parts. For instance, as we shall see later on, the existence of an Euler tour is very local, it depends only on the number of the neighbours of the vertices. Degrees of vertices DEFINITION. Let be a vertex a graph . The neighbourhood of is the set The degree of is the number of its neighbours: If , then is said to be isolated in , and if , then is a leaf of the graph. The minimum degree and the maximum degree of are defined as and The following lemma, due to EULER (1736), tells that if several people shake hands, then the number of hands shaken is even. Lemma 1.1 (Handshaking lemma). For each graph , Moreover, the number of vertices of odd degree is even. Proof. Every edge has two ends. The second claim follows immediately from the first one. Lemma 1.1 holds equally well for multigraphs, when is defined as the number of edges that have as an end, and when a loop is counted twice. Note that the degrees of a graph do not determine . Indeed, there are graphs and on the same set of vertices that are not isomorphic, but for which for all . 1.2 Subgraphs 8 DEFINITION. Let be a graph. A -switch of , for and , replaces the edges and by and . Before proving Berge’s switching theorem we need the following tool. Lemma 1.2. Let be a graph of order with a degree sequence , where . There is a graph which is obtained from by a sequence of -switches such that . Proof. Denote . Suppose that there exists a vertex with such that . Since , there exists a with such that . Here , since . Since , there exists a ( ) such that , but . We can now perform a -switch with respect to the vertices . This gives a new graph , where and , and the other neighbours of remain to be its neighbours. When we repeat this process for all indices with for , we obtain a graph as in the claim. Theorem 1.3 (BERGE (1973)). Two graphs and on a common vertex set satisfy for all if and only if can be obtained from by a sequence of -switches. Proof. If a graph is obtained from by a -switch, then clearly has the same degrees as . In the other direction, we use induction on the order . Let and have the same degrees, and let . By Lemma 1.2, there are sequences of -switches that transform to and to such that . Now the graphs and have the same degrees. By induction hypothesis, , and thus also , can be transformed to by a sequence of -switches. Finally, we observe that can be transformed to by the ‘inverse sequence’ of -switches, and this proves the claim. DEFINITION. Let be a descending sequence of nonnegative integers, that is, . Such a sequence is said to be graphical, if there exists a graph with such that for all . Using the next result recursively one can decide whether a sequence of integers is graphical or not. 1.2 Subgraphs 9 Theorem 1.4 (HAVEL (1955), HAKIMI (1962)). A sequence (with and ) is graphical if and only if (1.1) is graphical (when put into nonincreasing order). Proof. ( ) Consider of order with vertices (and degrees) as in (1.1). Add a new vertex and the edges for all . Then in this new graph , , and for all . ( ) Assume . By Lemma 1.2 and Theorem 1.3, we can suppose that . But now the degree sequence of is in (1.1). Example 1.3. Consider the sequence . By Theorem 1.4, is graphical is graphical is graphical is graphical The last sequence corresponds to a discrete graph , and hence also our original sequence is graphical. Indeed, the graph on the right has this degree sequence. Special graphs DEFINITION. A graph is trivial, if it has only one vertex, i.e., ; otherwise is nontrivial. The graph is the complete graph on , if every two vertices are adjacent: . All complete graphs of order are isomorphic with each other, and they will be denoted by . The complement of is the graph on , where . The complements of the complete graphs are called discrete graphs. In a discrete graph . Clearly, all discrete graphs of order are isomorphic with each other. A graph is said to be regular, if every vertex of has the same degree. If this degree is equal to , then is -regular or regular of degree . [...]... the graph, called the , where ắ ẫ if and only if the strings and differ in exactly Example 1. 5 Let For instance, -cube, with ẫ one place ẻ ẫ 11 0 ắ is ẫ , the number of binary strings of The order of is -regular, and so, by the handshaking length Also, lemma, ẫ Ă ẵ On the right we have the -cube, or simply the cube ẫ ắ 10 1 10 0 ề Example 1. 6 Let regular graph with shaking lemma 11 1 010 000 011 0 01. . .1. 2 Subgraphs 10 ề ẵà-regular In Note that a discrete graph is 0-regular, and a complete graph ề is for all graphs , and therefore particular, ề ềề ẵà ắ ềề ẵà ắ ề that have order Example 1. 4 The graph on the right is the Petersen graph that we will meet several times (drawn differently) It is a -regular graph of order ẵẳ ẵ be an integer, and consider... there exists a Notice that all -regular graphs have even order by the hand- ề ề is -regular Let be a -regular ắẹ ắ, and suppose that ắ ẻ ẻ ĩ í , and ề à ĩ ĩ í í ĩí Then is -regular of order ắẹ , then If graph of order Let í ĩ Subgraphs ẻ ẻ ẻ ẻ D EFINITION A graph is a subgraph of a graph , denoted by , if and A subgraph spans (and is a spanning subgraph of ), if every vertex of... , i.e., is an induced subgraph, if Also, a subgraph In this case, is induced by its set of vertices ẻ In an induced subgraph ẻ à , the set ắ ẻ à To each nonempty subset ẻ such that of edges consists of all ắ , there corresponds a unique induced subgraph àà 1. 3 Paths and cycles 11 To each subset of edges there corresponds a unique spanning subgraph of ẻ subgraph , à spanning induced... maximal complete subgraph (a subgraph ẹ of maximum order) of a graph is unlikely to be even in NP Reconstruction Problem The famous open problem, Kelly-Ulam problem or the Reconstruction Conjecture, states that a graph of order at least is determined up to isomorphism by its vertex deleted subgraphs ( ắ ): if there exists a bijection such for all , then that ôà ô ẻ ẻ 1. 3 Paths and cycles... every connected component is an induced subgraph, and ẻ ặ Ê à is the connected component of form a partition of that contains à ẵ ắ In particular, the connected components Shortest paths D EFINITION Let weight function ô ô be an edge weighted graph, that is, ấ on its edges For , let ôà ắ ẩ ô is a graph together with a ô à be the (total) weight of In particular, if is a path, then its weight... measure molecular attraction 1. 3 Paths and cycles 14 In these examples we look for a subgraph with the smallest weight, and which connects two given vertices, or all vertices (if we want to travel around) On the other hand, if the graph represents a network of pipelines, the weights are volumes or capacities, and then one wants to nd a subgraph with the maximum weight be a graph with an integer weight... fundamental notions in graph theory are practically oriented Indeed, many graph theoretical questions ask for optimal solutions to problems such as: nd a shortest path (in a complex network) from a given point to another This kind of problems can be difcult, or at least nontrivial, because there are usually choices what branch to choose when leaving an intermediate point 1. 3 Paths and cycles 12 Walks ắ be... be the subgraph of obtained by removing (only) the edges ắ from In particular, is obtained from by removing ắ , if each ắ (for is added to Similarly, we write For a subset that is, ẻ ã of vertices, we let be the subgraph induced by ẻ ề ẻ àà ẻ ề , and, e.g., is obtained from by removing the vertex together with the edges that have as their end Many problems concerning (induced) subgraphs are... convention A graph is connected, if à ẵ for all ắ ; otherwise, it is disconnected The maximal connected subgraphs of If à ẵ, then à are its connected components Denote the number of connected components of is, of course, connected The maximality condition means that a subgraph is a connected component if and is connected and there are no edges leaving , i.e., for every vertex ắ , the only if subgraph . the -cube, or simply the cube. 000 10 0 10 1 0 01 010 11 0 11 1 011 Example 1. 6. Let be any even number. We show by induction that there exists a - regular graph with . Notice that all -regular graphs. are the minimal weights from to each . 1. 3 Paths and cycles 15 The steps of the algorithm can also be rewritten as a table: 2 - - - - 3 3 3 - - 3 - - - 5 5 4 - 4 4 4 4 The correctness of Dijkstra’s. whether a sequence of integers is graphical or not. 1. 2 Subgraphs 9 Theorem 1. 4 (HAVEL (19 55), HAKIMI (19 62)). A sequence (with and ) is graphical if and only if (1. 1) is graphical (when put into nonincreasing