Graph Theory. Clemens Heuberger and Stephan Wagner. AIMS, January/February 2009


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1 Graph Theory Clemens Heuberger and Stephan Wagner AIMS, January/February Basic Definitions Definition 1.1 A graph is a pair G = (V (G),E(G)) of a set of vertices V (G) and a set of edges E(G), where the elements of E(G) are twoelement subsets of V (G). An oriented graph is a graph whose edges are oriented, i.e., each edge e has a distinguished head h(e) and a tail t(e). A directed graph is defined in a similar way, but a directed graph is additionally allowed to have an edge from a vertex x to a vertex y as well as an edge from y to x. A graph whose edges do not have a specific direction is also known as an undirected graph. The number of vertices of a graph G is called the order of G; the number of edges is called the size of G. A vertex v is incident with an edge e if v e. The two vertices v,w that are incident with an edge e are called the endvertices or ends of e; we write e = vw for short. Two vertices x,y of G are called adjacent or neighbours if xy is an edge of G; two edges are called adjacent if they have an end in common. Pairwise nonadjacent vertices or edges are called independent. A set of pairwise nonadjacent vertices or edges is thus called an independent set (stable set). We call two graphs G and H isomorphic (written as G H) if there exists a bijection φ : V (G) V (H) such that xy E(G) if and only if φ(x)φ(y) E(H). Such a map is called an isomorphism. If G = H, it is called an automorphism. If V (G) V (H) and E(G) E(H), we call G a subgraph of H (and H a supergraph of G); less formally, we say that H contains G. If E(G) contains all the edges of H between vertices of V (G), we say that G is an induced subgraph. Finally, if G is a subgraph of H and V (G) = V (H), we call G a spanning subgraph of H. The concept of a graph can be generalised in several ways; multigraphs are one such generalisation: unlike a (simple) graph, a multigraph is allowed to contain loops (a loop is an edge that connects a vertex with itself) and multiple edges (i.e. several copies of the same edge).
2 2 DEGREES AND DEGREE SEQUENCES 2 2 Degrees and Degree Sequences Definition 2.1 Let G be a graph and v a vertex of G. The neighbourhood N(v) of v is the set of all neighbours of v; the degree of v is the cardinality of N(v), i.e., the number of vertices of G that are adjacent to v: deg v = N(v). A vertex of degree 0 is called isolated. The number δ(g) = min{deg v : v V (G)} is the minimum degree of G, the number (G) = max{deg v : v V (G)} is the maximum degree of G. If all the vertices of G have degree k, we call G kregular (in particular, a 3regular graph is called cubic). Lemma 2.2 (Handshake Lemma) For any graph G, we have deg v = 2 E(G). v V (G) Proof: Every edge contributes exactly twice to the sum on the left hand side, once for each endpoint. Therefore, the sum must be equal to the number of edges, multiplied by 2. Definition 2.3 The degree sequence of a graph G is the sorted sequence of the degrees of all its vertices (in descending order). A sequence is called graphical if it is the degree sequence of some graph. Theorem 2.4 (HavelHakimi) A sequence a = (a 1,a 2,...,a k ) is graphical if and only if a 1 k 1 and a = (a 2 1,a 3 1,...,a a1 +1 1,a a1 +2,...,a k ) (possibly rearranged) is graphical. Proof: Suppose first that a is graphical, and that G is a graph whose vertex degrees are exactly given by this sequence. Let u 2,u 3,...,u a1 +1 be the vertices whose degrees are a 2 1,a 3 1,...,a a respectively. Add another vertex v, and connect it to all of the vertices u 2,u 3,...,u a1 +1. The degrees of all of these vertices increase by 1, and the newly introduced vertex v has degree a 1. Therefore, the degree sequence of the new graph is a, which implies that this sequence is also graphical. Conversely, assume that a is graphical, and that H is a graph whose vertex degrees are exactly given by this sequence. Let v be a vertex of degree a 1, and let u 2,u 3,...,u a1 +1 be the vertices whose degrees are a 2,a 3,...,a a1 +1 respectively. If u 2,u 3,...,u a1 +1 are the neighbours of v, we can simply remove v and obtain a graph whose degree sequence is a. Otherwise, let u i be a vertex that is not a neighbour of v (2 i a 1 + 1), and let u be a vertex that is a neighbour of v, but not among the vertices u 2,u 3,...,u a1 +1 (since v has a 1 neighbours, there must be such a vertex). Then, since a 1 a 2 a 3..., the degree of u must be smaller or equal to the degree of u i. v is a neighbour of u, but not of u i, thus there must be a neighbour v of u i that is not a neighbour of u. Now replace the edges vu and v u i by the edges vu i and v u, which does not change the degrees. We can repeat this transformation to obtain a new graph with the same vertex degrees and the property that the neighbours of v are precisely u 2,u 3,...,u a1 +1. Removing v now yields a graph with the desired degree sequence a.
3 3 PATHS, CYCLES, AND CONNECTIVITY 3 3 Paths, Cycles, and Connectivity Definition 3.1 A walk is an alternating sequence P = v 0,e 1,v 1,e 2,...,e k,v k of vertices and edges, beginning and ending with a vertex, where each vertex is incident to both the edge that precedes it and the edge that follows it in the sequence, and where the vertices that precede and follow an edge are the end vertices of that edge. Usually, we write P = v 0 v 1...v k for short and call P a walk from v 0 to v k (or v 0 v k walk for short). A walk is called closed if its first and last vertices are the same, and open if they are different. A path is an open walk in which no vertices are repeated, and a cycle is a closed walk in which no vertices are repeated. The length of a walk, path or cycle is the number of its edges. The distance d(v,w) of two vertices v,w is the length of a shortest vwpath; if no such path exists, we set d(v,w) =. The diameter of a graph G is the greatest distance between any two vertices of G. The minimum length of a cycle (if there are any) is called the girth, the maximum length is called the circumference. Theorem 3.2 Every graph contains a path of length δ(g) and, if δ(g) 2, a cycle of length at least δ(g) + 1. Proof: Let v 0 v 1...v k be a path of maximum length. Then all neighbours of v 0 must lie on the path, since we could otherwise extend the path by another edge. Since the degree of v 0 is at least δ(g), this implies k δ(g), i.e. the path has at least length δ(g). Furthermore, if δ(g) 2, then v 0 must have at least one neighbour on the path other than v 1. Let v i be the neighbour whose index i is maximal. Then, i deg v 0 δ(g), and v 0 v 1...v i v 0 is a cycle of length i + 1 δ(g) + 1. Definition 3.3 A nonempty graph G is called connected if any two of its vertices are linked by a path in G. G is called kconnected (for k N) if V (G) > k and G \ X (the graph that results from removing all vertices in X and all edges incident with these vertices) is connected for any vertex set X V (G) with X < k. The greatest integer k such that G is kconnected is called the connectivity κ(g) of G. Similarly, a graph G is called ledgeconnected if V (G) > 1 and G \ F (the graph that results from removing all edges in F) is connected for any edge set F E(G) with F < l. The greatest integer l such that G is ledgeconnected is the edgeconnectivity λ(g) of G. In particular, κ(g) = λ(g) = 0 if G is disconnected. Definition 3.4 A (connected) component of a graph G is a maximally connected subgraph (i.e., a connected subgraph of G that is not contained in any larger connected subgraph).
4 4 SPECIAL CLASSES OF GRAPHS 4 Theorem 3.5 For any graph G, we have κ(g) λ(g) δ(g). Proof: Let v be a vertex of minimum degree. By removing all edges incident with v, we can make the graph disconnected, and so we must have λ(g) δ(g). Now let e 1,e 2,...,e λ be a set of λ = λ(g) edges such that removing these edges yields a disconnected graph. The vertex set of this graph can thus be split into two parts V 1 and V 2 in such a way that there is no edge between V 1 and V 2. Furthermore, we know that each of the edges e 1,e 2,...,e λ has one endpoint in V 1 and another endpoint in V 2 (otherwise, we could save an edge and still obtain a disconnected graph, which is a contradiction to our choice of λ). If V 1 > λ, remove all the V 1 ends of the edges e 1,e 2,...,e λ. The resulting graph is disconnected (it still contains vertices of both V 1 and V 2, but no edges between them), and at most λ vertices have been removed. Therefore, κ(g) λ in this case. Otherwise, remove the V 1 ends of e 1,e 2,... until there is only one vertex of V 1 left. Then remove the V 2 ends of all the remaining edges in {e 1,e 2,...}. If there is still a vertex of V 2 left, then the remaining graph is disconnected, and at most λ vertices have been removed, so again κ(g) λ. If there is no vertex of V 2 left, then we know that the initial graph had no more than λ + 1 vertices, so that again κ(g) λ. 4 Special Classes of Graphs Definition 4.1 A complete graph of order n, denoted by K n, is a graph with n vertices and the property that any two vertices are connected by an edge. Definition 4.2 A kpartite graph is a graph whose vertex set V (G) can be partitioned into k classes V 1, V 2,..., V k such that there is no edge between vertices of the same class. In particular, a 2partite graph is called bipartite. A complete kpartite graph K n1,n 2,...,n k is a graph whose vertex set consists of k classes V 1, V 2,..., V k with V i = n i such that two vertices are connected by an edge if and only if they are in different classes. In particular, note that K n = K 1,1,...,1. Theorem 4.3 A graph G is bipartite if and only if it contains no odd cycle. Proof: Let first G be a bipartite graph with bipartition V 1,V 2 (i.e. V 1 V 2 = V (G), and all edges have one end in V 1 and the other in V 2 ). Suppose that v 1 v 2...v k v 1 is a cycle. Without loss of generality, we may assume that v 1 V 1. Then v 2 V 2, v 3 V 1, v 4 V 2, etc. In other words, the vertices with odd index lie in V 1 and those with even index lie in V 2. Since v k is a neighbour of v 1, we must have v k V 2, which implies that k is even. Thus every cycle has even length. Now suppose that G does not contain an odd cycle. We may assume that G is connected (otherwise, we can treat each connected component separately). Let v be a fixed vertex
5 4 SPECIAL CLASSES OF GRAPHS 5 and let V 1 be the set of vertices whose distance from v is odd and V 2 the set of vertices whose distance from v is even. We want to prove that there are no edges between vertices of V 1 and no edges between vertices of V 2. Assume that this is not the case: then there are vertices u 1 and u 2 which are connected by an edge such that there are a vu 1 path and a vu 2 path whose lengths are either both even or both odd. Taking the union of these two paths and the edge u 1 u 2, we obtain a closed walk of odd length. However, we still have to show that the existence of a closed walk of odd length implies the existence of a cycle of odd length. To this end, let v 1 v 2...v k v 1 be a closed walk of odd length k such that k is as small as possible. If this walk is not a cycle, then one vertex must be repeated, i.e. v i = v j for some i < j. But then one of the closed walks v 1 v 2...v i v j+1 v j+2...v k v 1 and v i v i+1...v j must have odd length, and this length is smaller than k, a contradiction. Therefore, the shortest closed walk of odd length must be an odd cycle. Definition 4.4 A forest is an acyclic graph, i.e., a graph that contains no cycles; a tree is a connected acyclic graph. Note that all connected components of a forest are trees. The vertices of degree 1 in a tree are called leaves. Lemma 4.5 Every tree with at least two vertices has at least two leaves. Proof: Let v 0 v 1...v k be a longest path of the tree. v 0 and v k cannot have neighbours outside the path, since the path could be extended further in this case. On the other hand, v 0 and v k cannot have neighbours on the path (other than v 1 and v k 1 respectively), since this would imply the existence of a cycle. Therefore, both v 0 and v k must be leaves. Theorem 4.6 The following statements are equivalent: 1. T is a tree; 2. T is minimally connected, i.e., it is connected, but T \ e is disconnected for any edge e E(T); 3. T is maximally acyclic, i.e., T is acyclic, but adding an edge to T always produces a cycle; Proof: First we show that 1. and 2. are equivalent. If T is a tree, then T is connected by definition; furthermore, if v and w are the ends of an edge e, then T \ e cannot contain a vwpath, since this path would form a cycle in T together with e. This implies that T \ e must be disconnected. Conversely, assume that 2. holds. We only have to show that T is acyclic. However, if T contained a cycle C, we could remove an edge e of C and still obtain a connected graph, contradicting 2. (if the path between two vertices in T contains the edge e, we can replace it by C \ e to obtain a connection in T \ e).
6 5 EULER WALKS AND TOURS 6 Next we show that 1. and 3. are equivalent. If T is a tree, then T is acyclic by definition. Suppose we add an edge e = vw to T that is not contained in T. Since T is connected, there must be a vwpath in T, and together with e, this path forms a cycle. Finally, assume that 3. holds. We have to show that T is connected. If not, we can choose vertices v and w that are not connected by a path. Add the edge e = vw to T. By our condition 3., this must produce a cycle (that contains e, since T itself is acyclic). Removing this edge from the cycle yields a path between v and w in T, a contradiction. Theorem 4.7 For any tree T, V (T) = E(T) + 1. Proof: Let V (T) = n; we use induction on n. For n = 1, the statement is obvious. If n 2, we know that there must be a leaf v. Remove this leaf to obtain another tree T = T \ v (T is a subgraph of T and thus still cyclefree; removing a leaf cannot result in a disconnected graph: a leaf can only be contained in a path between two vertices as one of its ends). By the induction hypothesis, it follows that V (T) = V (T ) + 1 = E(T ) + 2 = E(T) + 1, as desired. Theorem 4.8 Between any two vertices v,w of a tree T, there is a unique path in T. Proof: Since T is connected, there must be at least one such path. Suppose there are two different paths v 0 v 1...v k (v = v 0, w = v k ) and v 0v 1...v l (v = v 0, w = v l ) between v and w. Let i be minimal with the property that v i = v i and v i+1 v i+1 (such an index i must exist: at some point, the paths have to split); at some other vertex v j = v h, the paths have to meet again for the first time (at least, they have to meet at v k = v l ). Now it follows that v i v i+1...v j 1 v j v h 1v h 2...v i+1v i is a cycle, contradicting the definition of a tree. 5 Euler Walks and Tours Definition 5.1 An Euler walk of a graph G is a walk that traverses every edge of the graph exactly once. An Euler tour is a closed Euler walk. A graph G is called Eulerian if an Euler tour exists in G. Theorem 5.2 (Euler) A connected graph is Eulerian if and only if every vertex has even degree. A connected graph has an Euler walk if and only if all but at most two vertices have even degree. Proof: Let W be an Euler walk or tour starting at v and ending at w; each time the walk W passes a vertex u, exactly two edges are used: one to enter u and one to leave u. Since the Euler tour contains all edges exactly once, this implies that the number of edges
7 6 HAMILTONIAN PATHS AND CYCLES 7 incident with u must be even. The only possible exceptions are the vertices v and w; if v = w, however (so that the walk W is actually an Euler tour), the degree of v = w must also be even by the same argument. Conversely, assume that the graph G is connected and that every vertex has even degree. We want to show that G has an Euler tour (the proof for Eulerian walks is similar). Suppose that this is not the case and that G is a counterexample with as few edges as possible. Since there are no isolated vertices, all vertices must have degree 2. Start a walk at an arbitrary vertex that uses each edge at most once and continue it until a vertex is reached that has already been visited by the walk; since every vertex is incident with at least two edges, this is always possible. In this way, we obtain a closed walk that uses each edge at most once. Let W be the longest closed walk with this property, and assume that W is not yet an Euler tour. If we remove W, we obtain a graph G = G \ W with the property that all vertices have even degree. Since W is not yet an Euler tour, G must contain some edges, and so there is a connected component of G that has an Euler tour T (because of the assumption that G was a counterexample with minimum number of edges). Since G is connected, W and T must have a common vertex. Now we can join W and T to form a closed walk W that uses each edge at most once. But W is longer than W, which contradicts the choice of W. Therefore, W must be an Euler tour. 6 Hamiltonian Paths and Cycles Definition 6.1 A Hamiltonian path is a path that contains every vertex of a graph G; a Hamiltonian cycle is a cycle that contains every vertex. If G has a Hamiltonian cycle, it is called Hamiltonian. Theorem 6.2 (Dirac) Every graph G with n 3 vertices and minimum degree δ(g) n/2 is Hamiltonian. Proof: Let v 1 v 2...v k be a path of maximum length in G; clearly, k n. All neighbours of v 1 and v k must lie on this path; otherwise, we could extend the path by another edge. Let and I = {i : v i+1 is a neighbour of v 1 } J = {i : v i is a neighbour of v k }. Then both I and J are subsets of {1, 2,...,k 1}, and both I and J contain at least δ(g) n/2 > (k 1)/2 elements. Therefore, I and J must have a common element i. Now note that C = v 1 v 2...v i v k v k 1...v i+1 v 1 is a cycle. If this cycle is a Hamiltonian cycle, we are done. Otherwise, there must be vertices of G not contained in C, and since G is connected, there must be a vertex w that
8 7 PLANAR GRAPHS 8 is adjacent to a vertex v j contained in C; if we add the edge v j w and remove one of the edges incident with v j from C, we obtain a new path that is longer than the initial path v 1 v 2...v k, which contradicts the choice of this path. Therefore, C must be a Hamiltonian cycle. 7 Planar Graphs Definition 7.1 A plane graph is a graph whose vertices are points in the plane and whose edges are represented by curves between the respective endpoints in such a way that edges intersect only at their endpoints. A planar graph is a graph that can be embedded in the plane in this way, i.e., a graph that is isomorphic to a plane graph. The regions bounded by the edges of a plane graph are called faces. Theorem 7.2 If G is a plane graph with n vertices, m edges, c connected components and l faces, then n m + l = c + 1. Proof: We use induction on the number of edges. If m = 0, then there are n isolated vertices (and thus c = n) and exactly one face (l = 1). The stated formula obviously holds. If an edge is added, we have two possibilities: An edge is added between two vertices that are already connected. In this case, a face is split in two parts, thus increasing l by 1. Since m increases by 1 and n and c remain the same, the formula still holds. An edge is added between two connected components. In this case, the number of faces stays the same, but the number of connected components decreases by 1. Again, the stated formula remains correct. Corollary 7.3 (Euler s formula) Let G be a connected plane graph with n vertices, m edges and l faces. Then n m + l = 2. Corollary 7.4 A planar graph with n 3 vertices has at most 3n 6 edges. Proof: Note that every face has at least three edges on its boundary (unless the graph has less than three edges), and every edge is only part of the boundary of at most two faces. Therefore, if m denotes the number of edges and l the number of faces, we have Using this in Euler s formula, we obtain 3l 2m. 2 = n m + l n m + 2m 3 = n m 3, which can be rewritten as m 3n 6.
9 7 PLANAR GRAPHS 9 Corollary 7.5 The graphs K 5 and K 3,3 are not planar. Proof: Note that the complete graph K 5 has five vertices and 10 > 9 = edges, contradicting the inequality above. The proof that K 3,3 is not planar uses the same idea and is left as an exercise. Theorem 7.6 (Kuratowski) A finite graph is planar if and only if it does not contain a subgraph that is a subdivision (i.e., results from inserting additional vertices on the edges) of K 5 or K 3,3.
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