Graph Theory Ch. 1. Fundamental Concept 1 Chapter 1 Fundamental Concept 1.1 What Is a Graph? 1.2 Paths, Cycles, and Trails 1.3 Vertex Degree and Counting 1.4 Directed Graphs

Graph Theory Ch. 1. Fundamental Concept 2 The K Ö nigsberg Bridge Problem  Königsber is a city on the Pregel river in Prussia  The city occupied two islands plus areas on both banks  Problem: Whether they could leave home, cross every bridge exactly once, and return home. X Y Z W

Graph Theory Ch. 1. Fundamental Concept 3 A Model  A vertex: a region  An edge: a path(bridge) between two regions e1e1 e2e2 e3e3 e4e4 e6e6 e5e5 e7e7 Z Y X W X Y Z W

Graph Theory Ch. 1. Fundamental Concept 4 What Is a Graph?  A graph G is a triple consisting of: –A vertex set V(G) –An edge set E(G) –A relation between an edge and a pair of vertices e1e1 e2e2 e3e3 e4e4 e6e6 e5e5 e7e7 Z Y X W

Graph Theory Ch. 1. Fundamental Concept 5 Loop, Multiple edges  Loop: An edge whose endpoints are equal  Multiple edges: Edges have the same pair of endpoints loop Multiple edges

Graph Theory Ch. 1. Fundamental Concept 6 Simple Graph  Simple graph: A graph has no loops or multiple edges loop Multiple edges It is not simple. It is a simple graph.

Graph Theory Ch. 1. Fundamental Concept 7 Adjacent, neighbors  Two vertices are adjacent and are neighbors if they are the endpoints of an edge.  Example: –A and B are adjacent. –A and D are not adjacent. A B C D

Graph Theory Ch. 1. Fundamental Concept 8 Finite Graph, Null Graph  Finite graph: an graph whose vertex set and edge set are finite.  Null graph: the graph whose vertex set and edges are empty.

Graph Theory Ch. 1. Fundamental Concept 9 Complement  Complement of G: The complement G’ of a simple graph G : –A simple graph –V(G’) = V(G) –E(G’) = { uv | uv  E(G) } G u v w x y G’ u v w x y

Graph Theory Ch. 1. Fundamental Concept 10 Clique and Independent set  A Clique in a graph: a set of pairwise adjacent vertices (a complete subgraph)  An independent set in a graph: a set of pairwise nonadjacent vertices.  Example: –{x, y, u} is a clique in G. –{u, w} is an independent set. G u v w x y

Graph Theory Ch. 1. Fundamental Concept 11 Bipartite Graphs  A graph G is bipartite if V(G) is the union of two disjoint independent sets called partite sets of G  Also: The vertices can be partitioned into two sets such that each set is independent  Matching Problem  Job Assignment Problem Workers Jobs Boys Girls

Graph Theory Ch. 1. Fundamental Concept 12 Chromatic Number  The chromatic number of a graph G, written x(G), is the minimum number of colors needed to label the vertices so that adjacent vertices receive different colors Red Green Blue x(G) = 3

Graph Theory Ch. 1. Fundamental Concept 13 Maps and coloring  A map is a partition of the plane into connected regions  Can we color the regions of every map using at most four colors so that neighboring regions have different colors?  Map Coloring  graph coloring –A region  A vertex –Adjacency  An edge

Graph Theory Ch. 1. Fundamental Concept 14 Scheduling and graph Coloring 1  Two committees can not hold meetings at the same time if two committees have common member common member Committee 1 Committee 2

Graph Theory Ch. 1. Fundamental Concept 15 Scheduling and graph Coloring 1  Model: –One committee being represented by a vertex –An edge between two vertices if two corresponding committees have common member –Two adjacent vertices can not receive the same color common member Committee 1 Committee 2

Graph Theory Ch. 1. Fundamental Concept 16 Scheduling and graph Coloring 2  Scheduling problem is equivalent to graph coloring problem. common member Committee 1 Committee 2 Committee 3 Common Member Different Color No Common Member Same Color OK Same time slot OK

Graph Theory Ch. 1. Fundamental Concept 17 Path and Cycle  Path: a sequence of distinct vertices such that two consecutive vertices are adjacent. –Example: (a, d, c, b, e) is a path –(a, b, e, d, c, b, e, d) is not a path; it is a walk.  Cycle: a closed Path –Example: (a, d, c, b, e, a) is a cycle a b c d e

Graph Theory Ch. 1. Fundamental Concept 18 Subgraphs  A subgraph of a graph G is a graph H such that: –V(H)  V(G) and E(H)  E(G) and –The assignment of endpoints to edges in H is the same as in G.

Graph Theory Ch. 1. Fundamental Concept 19 Subgraphs  Example: H 1, H 2, and H 3 are subgraphs of G c d a b d e a b c d e H1H1 G H3H3 H2H2 a b c d e

Graph Theory Ch. 1. Fundamental Concept 20 Connected and Disconnected  Connected: There exists at least one path between two vertices.  Disconnected: Otherwise  Example: –H 1 and H 2 are connected. –H 3 is disconnected. c d a b d e a b c d e H1H1 H3H3 H2H2

Graph Theory Ch. 1. Fundamental Concept 21 Adjacency, Incidence, and Degree  Assume e i is an edge whose endpoints are (v j,v k )  The vertices v j and v k are said to be adjacent.  The edge e i is said to be incident upon v j  Degree of a vertex v k is the number of edges incident upon v k. It is denoted as d(v k ) eiei vjvj vkvk

Graph Theory Ch. 1. Fundamental Concept 22 Adjacency matrix  Let G = (V, E), |V| = n and |E|=m  The adjacency matrix of G written A(G), is the n- by-n matrix in which entry a i,j is the number of edges in G with endpoints {v i, v j }. a b c d e w x y z w x y z wxyzwxyz

Graph Theory Ch. 1. Fundamental Concept 23 Incidence Matrix  Let G = (V, E), |V| = n and |E|=m  The incidence matrix M(G) is the n-by-m matrix in which entry m i,j is 1 if v i is an endpoint of e i and otherwise is 0. a b c d e w x y z a b c d e wxyzwxyz

Graph Theory Ch. 1. Fundamental Concept 24 Isomorphism  An isomorphism from a simple graph G to a simple graph H is a bijection f:V(G)  V(H) such that uv  E(G) if and only if f(u)f(v)  E(H) –We say “G is isomorphic to H”, written G  H H G w x z y c d b a f 1 : w x y z c b d a f 2 : w x y z a d b c

Graph Theory Ch. 1. Fundamental Concept 25 Complete Graph, Complete Bipartite Graph or Biclique  Complete Graph: a simple graph whose vertices are pairwise adjacent.  Complete bipartite graph (biclique) is a simple bipartite graph such that two vertices are adjacent if and only if they are in different partite sets. Complete Graph Complete Bipartite Graph

Graph Theory Ch. 1. Fundamental Concept 26 Petersen Graph  The petersen graph is the simple graph whose vertices are the 2-element subsets of a 5-element set and whose edges are pairs of disjoint 2-element subsets

Graph Theory Ch. 1. Fundamental Concept 27 Petersen Graph  Assume: the set of 5-element be (1, 2, 3, 4, 5) –Then, 2-element subsets: (1,2) (1,3) (1,4) (1,5) (2,3) (2,4) (2,5) (3,4) (3,5) (4,5) Disjoint, so connected 45: (4, 5)

Graph Theory Ch. 1. Fundamental Concept 28 Petersen Graph  Three drawings

Graph Theory Ch. 1. Fundamental Concept 29 Theorem: If two vertices are non-adjacent in the Petersen Graph, then they have exactly one common neighbor Proof: x, z x, y No connection, Joint, One common element. u, v Since 5 elements totally, 5-3 elements left. Hence, exactly one of this kind. 3 elements in these vertices totally

Graph Theory Ch. 1. Fundamental Concept 30 Girth ,  Girth: the length of its shortest cycle. –If no cycles, girth is infinite

Graph Theory Ch. 1. Fundamental Concept 31 Girth and Petersen graph ,  Theorem: The Petersen Graph has girth 5. Proof: –Simple  no loop  no 1-cycle (cycle of length 1) –Simple  no multiple  no 2-cycle –5 elements  no three pair-disjoint 2-sets  no 3- cycle –By previous theorem, two nonadjacent vertices has exactly one common neighbor  no 4-cycle – is a 5-cycle.

Graph Theory Ch. 1. Fundamental Concept 32 Walks, Trails  A walk: a list of vertices and edges v 0, e 1, v 1, …., e k, v k such that, for 1  i  k, the edge e i has endpoints v i-1 and v i.  A trail : a walk with no repeated edge.

Graph Theory Ch. 1. Fundamental Concept 33 Paths  A u,v-walk or u,v-trail has first vertex u and last vertex v; these are its endpoints.  A u,v-path: a u,v-trail with no repeated vertex.  The length of a walk, trail, path, or cycle is its number of edges.  A walk or trail is closed if its endpoints are the same.

Graph Theory Ch. 1. Fundamental Concept 34 Lemma: Every u,v-walk contains a u,v-path Proof:  Use induction on the length l of a u, v-walk W.  Basis step: l = 0. –Having no edge, W consists of a single vertex (u=v). –This vertex is a u,v-path of length 0. to be continued

Graph Theory Ch. 1. Fundamental Concept 35 Lemma: Every u,v-walk contains a u,v-path Proof: Continue  Induction step : l  1. (see the figure in the next page) –Suppose that the claim holds for walks of length less than l. –If W has no repeated vertex, then its vertices and edges form a u,v-path. –If W has a repeated vertex w, then deleting the edges and vertices between appearances of w (leaving one copy of w) yields a shorter u,v-walk W’ contained in W. (see next page) –By the induction hypothesis, W ’ contains a u,v- path P,and this path P is contained in W.

Graph Theory Ch. 1. Fundamental Concept 36 Lemma: Every u,v-walk contains a u,v-path  An example: Delete

Graph Theory Ch. 1. Fundamental Concept 37 Components  The components of a graph G are its maximal connected subgraphs.  A component (or graph) is trivial if it has no edges; otherwise it is nontrivial.  An isolated vertex is a vertex of degree 0.

Graph Theory Ch. 1. Fundamental Concept 38 Proof: –An n-vertex graph with no edges has n components –Each edge added reduces this by at most 1 –If k edges are added, then the number of components is at least n-k Theorem: Every graph with n vertices and k edges has at least n-k components

Graph Theory Ch. 1. Fundamental Concept 39 Theorem: Every graph with n vertices and k edges has at least n-k components  Examples: n=2, k=1, 1 component n=3, k=2, 1 component n=6, k=3, 3 components n=6, k=3, 4 components

Graph Theory Ch. 1. Fundamental Concept 40 Cut-edge, Cut-vertex  A cut-edge or cut-vertex of a graph is an edge or vertex whose deletion increases the number of components. Not a Cut-vertex Cut-edge Cut-vertex

Graph Theory Ch. 1. Fundamental Concept 41 Cut-edge, Cut-vertex  G-e or G-M : The subgraph obtained by deleting an edge e or set of edges M.  G-v or G-S : The subgraph obtained by deleting a vertex v or set of vertices S. e G-eG-e G

Graph Theory Ch. 1. Fundamental Concept 42 Induced subgraph  An induced subgraph: –A subgraph obtained by deleting a set of vertices. –We write G [ T ] for G - T ’, where T ’ =V( G )- T ; –G [ T ] is the subgraph of G induced by T.  Example: –Assume T:{A, B, C, D} B A C D E B A C D G[T]G[T] G

Graph Theory Ch. 1. Fundamental Concept 43 Induced subgraph  More Examples: –G 2 is the subgraph of G 1 induced by (A, B, C, D) –G 3 is the subgraph of G 1 induced by (B, C) –G 4 is not the subgraph induced by (A, B, C, D) B A C D E B A C D B C B A C D G1G1 G2G2 G3 G4G4

Graph Theory Ch. 1. Fundamental Concept 44 Induced subgraph  A set S of vertices is an independent set if and only if the subgraph induced by it has no edges. –G 3 is an example. B A C D E B C G1G1 G3

Graph Theory Ch. 1. Fundamental Concept 45 Theorem: An edge e is a cut-edge if and only if e belongs to no cycles Proof : 1/2  Let e= (x, y) be an edge in a graph G and H be the component containing e. –Since deletion of e effects no other component, it suffices to prove that H-e is connected if and only if e belongs to a cycle.  First suppose that H-e is connected. –This implies that H-e contains an x, y-path, –This path completes a cycle with e.

Graph Theory Ch. 1. Fundamental Concept 46 Theorem: An edge e is a cut-edge if and only if e belongs to no cycles Proof : 2/2  Now suppose that e lies in a cycle C. –Choose u, v  V(H) Since H is connected, H has a u, v-path P. –If P does not contain e Then P exists in H-e –Otherwise Suppose by symmetry that x is between u and y on P Since H-e contains a u, x-path along P, the transitivity of the connection relation implies that H-e has a u, v-path. –We did this for all u, v  V(H), so H-e is connected.

Graph Theory Ch. 1. Fundamental Concept 47 Theorem: An edge e is a cut-edge if and only if e belongs to no cycles  An Example:

Graph Theory Ch. 1. Fundamental Concept 48 Lemma: Every closed odd walk contains an odd cycle Proof: 1/2  Use induction on the length l of a closed odd walk W.  l=1. A closed walk of length 1 traverses a cycle of length 1.  We need to prove the claim holds if it holds for closed odd walks shorter than W.

Graph Theory Ch. 1. Fundamental Concept 49 Lemma: Every closed odd walk contains an odd cycle Proof: 2/2  Suppose that the claim holds for closed odd walks shorter than W. –If W has no repeated vertex (other than first = last), then W itself forms a cycle of odd length. –Otherwise, Need to prove: If repeated, W includes a shorter closed odd walk. By induction, the theorem hold If W has a repeated vertex v, then we view W as starting at v and break W into two v,v-walks. Since W has odd length, one of these is odd and the other is even. (see the next page) The odd one is shorter than W, by induction hypothesis, it contains an odd cycle, and this cycle appears in order in W.

Graph Theory Ch. 1. Fundamental Concept 50 Lemma: Every closed odd walk contains an odd cycle Even v Odd Odd = Odd + Even

Graph Theory Ch. 1. Fundamental Concept 51 Theorem: A graph is bipartite if and only if it has no odd cycle  Examples: B A D C A C B D A C B D F E A C B D E F

Graph Theory Ch. 1. Fundamental Concept 52 Theorem: A graph is bipartite if it has no odd cycle Proof: (sufficiency 1/3 )  Let G be a graph with no odd cycle.  We prove that G is bipartite by constructing a bipartition of each nontrivial component H.  For each v  V(H), let f(v) be the minimum length of a u, v- path. Since H is connected, f(v) is defined for each v  V(H).

Graph Theory Ch. 1. Fundamental Concept 53 Theorem: A graph is bipartite if it has no odd cycle Proof: (sufficiency 2/3 )  Let X={v  V(H): f(v) is even} and Y={v  V(H): f(v) is odd}  An edge v,v’ within X (or Y) would create a closed odd walk using a shortest u, v-path, the edge v, v’ within X (or Y) and the reverse of a shortest u, v’-path. A closed odd walk using 1)a shortest u, v-path, 2)the edge v, v’ within X (or Y), and 3)the reverse of a shortest u, v’-path.

Graph Theory Ch. 1. Fundamental Concept 54 Theorem: A graph is bipartite if it has no odd cycle Proof: (sufficiency 3/3 )  By Lemma , such a walk must contain an odd cycle, which contradicts our hypothesis  Hence X and Y are independent sets. Also X  Y = V(H), so H is an X, Y-bipartite graph Even (or Odd) Odd Cycle Because: even (or odd) + even (or odd) = even even + 1 = odd Since no odd cycles, vv’ doesn’t exist. We have: X and Y are independent sets

Graph Theory Ch. 1. Fundamental Concept 55 Theorem: A graph is bipartite only if it has no odd cycle Proof: (necessity)  Let G be a bipartite graph.  Every walk alternates between the two sets of a bipartition  So every return to the original partite set happens after an even number of steps  Hence G has no odd cycle

Graph Theory Ch. 1. Fundamental Concept 56 Eulerian Circuits  A graph is Eulerian if it has a closed trail containing all edges.  We call a closed trail a circuit when we do not specify the first vertex but keep the list in cyclic order.  An Eulerian circuit or Eulerian trail in a graph is a circuit or trail containing all the edges.

Graph Theory Ch. 1. Fundamental Concept 57 Even Graph, Even Vertex, and Maximal Path  An even graph is a graph with vertex degrees all even.  A vertex is odd [even] when its degree is odd [even].  A maximal path in a graph G is a path P in G that is not contained in a longer path. –When a graph is finite, no path can extend forever, so maximal (non-extendible) paths exist.

Graph Theory Ch. 1. Fundamental Concept 58 Lemma: If every vertex of graph G has degree at least 2, then G contains a cycle Proof:  Let P be a maximal path in G, and let u be an endpoint of P  Since P cannot be extended, every neighbor of u must already be a vertex of P  Since u has degree at least 2, it has a neighbor v in V(P) via an edge not in P  The edge uv completes a cycle with the portion of P from v to u u P Impossible v P u Must

Graph Theory Ch. 1. Fundamental Concept 59 Theorem: A graph G is Eulerian if and only if it has at most one nontrivial component and its vertices all have even degree Proof: (Necessity)  Suppose that G has an Eulerian circuit C.  Each passage of C through a vertex uses two incident edges, and the first edge is paired with the last at the first vertex.  Hence every vertex has even degree. Also, two edges can be in the same trail only when they lie in the same component, so there is at most one nontrivial component.

Graph Theory Ch. 1. Fundamental Concept 60 Theorem: A graph G is Eulerian if and only if it has at most one nontrivial component and its vertices all have even degree Proof: (Sufficiency 1/3 )  Assuming that the condition holds, we obtain an Eulerian circuit using induction on the number of edges, m.  Basis step: m=0. A closed trail consisting of one vertex suffices. →

Graph Theory Ch. 1. Fundamental Concept 61 Theorem: A graph G is Eulerian if and only if it has at most one nontrivial component and its vertices all have even degree Proof: (Sufficiency 2/3 )  Induction step: m>0. –When even degrees, each vertex in the nontrivial component of G has degree at least 2. –By Lemma , the nontrivial component has a cycle C. – Let G’ be the graph obtained from G by deleting E(C). –Since C has 0 or 2 edges at each vertex, each component of G’ is also an even graph. –Since each component is also connected and has fewer than m edges, we can apply the induction hypothesis to conclude that each component of G’ has an Eulerian circuit. →

Graph Theory Ch. 1. Fundamental Concept 62 Theorem: A graph G is Eulerian if and only if it has at most one nontrivial component and its vertices all have even degree Proof: (Sufficiency 3/3 )  Induction step: m>0. (continued) –To combine these into an Eulerian circuit of G, we traverse C, but when a component of G’ is entered for the first time we detour along an Eulerian circuit of that component. –This circuit ends at the vertex where we began the detour. When we complete the traversal of C, we have completed an Eulerian circuit of G.

Graph Theory Ch. 1. Fundamental Concept 63 Proposition: Every even graph decomposes into cycles Proof:  In the proof of Theorem –It is noted that every even nontrivial graph has a cycle –The deletion of a cycle leaves an even graph  Thus this proposition follows by induction on the number of edges

Graph Theory Ch. 1. Fundamental Concept 64 Proposition: If G is a simple graph in which every vertex has degree at least k, then G contains a path of length at least k. If k  2, then G also contains a cycle of length at least k Proof: (1/2)  Let u be an endpoint of a maximal path P in G.  Since P does not extend, every neighbor of u is in V(P).  Since u has at least k neighbors and G is simple, P therefore has at least k vertices other than u and has length at least k.

Graph Theory Ch. 1. Fundamental Concept 65 Proposition: If G is a simple graph in which every vertex has degree at least k, then G contains a path of length at least k. If k  2, then G also contains a cycle of length at least k Proof: (2/2)  If k  2, then the edge from u to its farthest neighbor v along P completes a sufficiently long cycle with the portion of P from v to u. u v d(u)  k At least k+1 vertices Length  k

Graph Theory Ch. 1. Fundamental Concept 66 Degree  The degree of vertex v in a graph G, written or d(v), is the number of edges incident to v, except that each loop at v counts twice  The maximal degree is  (G)  The minimum degree is  (G) A C B D F E d(B) = 3, d(C) = 2 Δ(G) = 3, δ(G) = 2 G

Graph Theory Ch. 1. Fundamental Concept 67 Regular  G is regular if  (G) =  (G)  G is k-regular if the common degree is k.  The neighborhood of v, written N g (v) or N(v) is the set of vertices adjacent to v. 3-regular

Graph Theory Ch. 1. Fundamental Concept 68 Order and size  The order of a graph G, written n(G), is the number of vertices in G.  An n-vertex graph is a graph of order n.  The size of a graph G, written e(G), is the number of edges in G.  For n  N, the notation [n] indicates the set {1,…, n}.

Graph Theory Ch. 1. Fundamental Concept 69 Proposition: (Degree-Sum Formula) If G is a graph, then  v  V(G) d(v) = 2 e(G) Proof:  Summing the degrees counts each edge twice, – Because each edge has two ends and contributes to the degree at each endpoint.

Graph Theory Ch. 1. Fundamental Concept 70 Theorem: If k>0, then a k-regular bipartite graph has the same number of vertices in each partite set Proof:  Let G be an X,Y-bigraph.  Counting the edges according to their endpoints in X yields e(G)=k|X|. d (x) = k x

Graph Theory Ch. 1. Fundamental Concept 71 Theorem: If k>0, then a k-regular bipartite graph has the same number of vertices in each partite set Proof:  Counting them by their endpoints in Y yields e(G)=k|Y|.  Thus k|X|=k|Y|, which yields |X|=|Y| when k>0. d (x) = k x d (y) = k y

Graph Theory Ch. 1. Fundamental Concept 72 A technique for counting a set 1/  Example: The Petersen graph has ten 6-cycles –Let G be the Petersen graph. –Being 3-regular, G has ten copies of K 1,3 (claw). We establish a one-to-one correspondence between the 6- cycles and the claws. –Since G has girth 5, every 6-cycle F is an induced subgraph. see below –Each vertex of F has one neighbor outside F. d(v)= 3, v  V(G) If Existing, Girth =3. But Girth=5 so no such an edge

Graph Theory Ch. 1. Fundamental Concept 73 A technique for counting a set 2/ –Since nonadjacent vertices have exactly one common neighbor (Proposition ), opposite vertices on F have a common neighbor outside F. –Since G is 3-regular, the resulting three vertices outside F are distinct. –Thus deleting V(F) leaves a subgraph with three vertices of degree 1 and one vertex of degree 3; it is a claw. Common neighbor of opposite vertices If the neighbors are not distinct, d(v)>3

Graph Theory Ch. 1. Fundamental Concept 74 A technique for counting a set 3/ –It is shown that each claw H in G arises exactly once in this way. –Let S be the set of vertices with degree 1 in H; S is an independent set. –The central vertex of H is already a common neighbor, so the six other edges from S reach distinct vertices. –Thus G-V(H) is 2-regular. Since G has girth 5, G-V(H) must be a 6-cycle. This 6-cycle yields H when its vertices are deleted.

Graph Theory Ch. 1. Fundamental Concept 75 Proposition: The minimum number of edges in a connected graph with n vertices is n Proof:  By proposition , every graph with n vertices and k edges has at least n-k components.  Hence every n-vertex graph with fewer than n-1 edges has at least two components and is disconnected.  The contrapositive of this is that every connected n-vertex graph has at least n- 1 edges. This lower bound is achieved by the path P n.

Graph Theory Ch. 1. Fundamental Concept 76 Theorem: If G is simple n-vertex graph with  (G)  (n- 1 )/ 2, then G is connected Proof: 1/2  Choose u,v  V(G).  It suffices to show that u,v have a common neighbor if they are not adjacent.  Since G is simple, we have |N(u) |   (G)  (n-1)/2, and similarly for v. –Recall:  (G) is the minimum degree, |N(u)| = d(u) Hence: |N(u) |   (G)

Graph Theory Ch. 1. Fundamental Concept 77 Theorem: If G is simple n-vertex graph with  (G)  (n-1)/2, then G is connected Proof: 2/2  When u and v are not connected, we have |N(u)  N(v) |  n-2 – since u and v are not in the union  Using Remark A.13 of Appendix A, we thus compute

Graph Theory Ch. 1. Fundamental Concept 78 Theorem: Every loopless graph G has a bipartite subgraph with at least e(G)/2 edges Proof:  Partition V(G) into two sets X, Y.  Using the edges having one endpoint in each set yields a bipartite subgraph H with bipartition X, Y.  If H contains fewer than half the edges of G incident to a vertex v, then v has more edges to vertices in its own class than in the other class, as illustrated bellow.

Graph Theory Ch. 1. Fundamental Concept 79 Proof: 2/2  Moving v to the other class gains more edges of G than it loses.  Using Iterative improvement approach  When it terminates, we have d H (v)  d G (v)/2 for every v  V(G).  Summing this and applying the degree-sum formula yields e(H)  e(G)/2. Theorem: Every loopless graph G has a bipartite subgraph with at least e(G)/2 edges

Graph Theory Ch. 1. Fundamental Concept 80 Example  The algorithm in Theorem need not produce a bipartite subgraph with the most edges, merely one with at least half the edges. - Local Maximum.  Consider the graph in the next page. –It is 5-regular with 8 vertices and hence has 20 edges. –The bipartition X={a,b,c,d} and Y={e,f,g,h} yields a 3-regular bipartite subgraph with 12 edges. –The algorithm terminates here; switching one vertex would pick up two edges but lose three.

Graph Theory Ch. 1. Fundamental Concept 81 Example(Cont.)

Graph Theory Ch. 1. Fundamental Concept 82 Example  Nevertheless, the bipartition X={a,b,g,h} and Y={c,d,e,f} yields a 4-regular bipartite subgraph with 16 edges.  An algorithm seeking the maximal by local changes may get stuck in a local maximum. Local Maximum Global Maximum

Graph Theory Ch. 1. Fundamental Concept 83 Theorem: The maximum number of edges in an n- vertex triangle free simple graph is  n 2 / 4  Proof : 1/6 –Let G be an n-vertex triangle-free simple graph. –Let x be a vertex of maximum degree and d(x)=k. –Since G has no triangles, there are no edges among neighbors of x. No edges between neighbors of x

Graph Theory Ch. 1. Fundamental Concept 84 Theorem: The maximum number of edges in an n- vertex triangle free simple graph is  n 2 / 4  Proof : 2/6 –Hence summing the degrees of x and its nonneighbors counts at least one endpoint of every edge:  v  N(x) d(v)  e(G). –We sum over n-k vertices, each having degree at most k, so e(G)  (n-k)k

Graph Theory Ch. 1. Fundamental Concept 85 Doesn’t exist  v  N(x) d(v) counts at least one endpoint of every edge At most k vertices At least n-k vertices No edges exist Theorem: The maximum number of edges in an n- vertex triangle free simple graph is  n 2 / 4  Proof: 3/6

Graph Theory Ch. 1. Fundamental Concept 86 Proof: 4/6 –Since (n-k)k counts the edges in K n-k, k, we have now proved that e(G) is bounded by the size of some biclique with n vertices. i.e. e(G)  (n-k)k = |the edges in K n-k, k | Theorem: The maximum number of edges in an n- vertex triangle free simple graph is  n 2 / 4  n-k k

Graph Theory Ch. 1. Fundamental Concept 87 Proof: 5/6 –Moving a vertex of K n-k,k from the set of size k to the set of size n-k gains k-1 edges and loses n-k edges. –The net gain is 2k-1-n, which is positive for 2k>n+1 and negative for 2k<n+1. –Thus e(K n-k, k ) is maximized when k is  n/ 2  or  n/ 2 . Theorem: The maximum number of edges in an n- vertex triangle free simple graph is  n 2 / 4  n-k k

Graph Theory Ch. 1. Fundamental Concept 88 Proof: 6/6 –The product is then n 2 /4 for even n and (n )/4 for odd n. Thus e(G)   n 2 /4 . –The bound is best possible. It is seen that a triangle-free graph with  n 2 /4  edges is: K  n/2 ,  n/2 . Theorem: The maximum number of edges in an n- vertex triangle free simple graph is  n 2 / 4 

Graph Theory Ch. 1. Fundamental Concept 89 Degree sequence  The Degree Sequence of a graph is the list of vertex degrees, usually written in non-increasing order, as d 1  ….  d n.  Example: z y x w v Degree sequence: d(w), d(x), d(y), d(z), d(v)

Graph Theory Ch. 1. Fundamental Concept 90 Proposition: The nonnegative integers d 1,…, d n are the vertex degrees of some graph if and only if  d i is even Proof: ½ Necessity  When some graph G has these numbers as its vertex degrees, the degree-sum formula implies that  d i = 2e (G), which is even.

Graph Theory Ch. 1. Fundamental Concept 91 Proof: 2/2 Sufficiency  Suppose that  d i is even.  We construct a graph with vertex set v 1,…,v n and d(v i ) = d i for all i.  Since  d i is even, the number of odd values is even.  First form an arbitrary pairing of the vertices in {v i : d i is odd}.  For each resulting pair, form an edge having these two vertices as its endpoints  The remaining degree needed at each vertex is even and nonnegative; satisfy this for each i by placing [d i /2] loops at v i Proposition: The nonnegative integers d 1,…, d n are the vertex degrees of some graph if and only if  d i is even

Graph Theory Ch. 1. Fundamental Concept 92 Graphic Sequence  A graphic sequence is a list of nonnegative numbers that is the degree sequence of some simple graph.  A simple graph “realizes” d. –means: A simple graph with degree sequence d.

Graph Theory Ch. 1. Fundamental Concept 93 Recursive condition  The lists (2, 2, 1, 1) and (1, 0, 1) are graphic. The graphic K 2 +K 1 realizes 1, 0, 1.  Adding a new vertex adjacent to vertices of degrees 1 and 0 yields a graph with degree sequence 2, 2, 1, 1, as shown below.  Conversely, if a graph realizing 2, 2, 1, 1 has a vertex w with neighbors of degrees 2 and 1, then deleting w yields a graph with degrees 1, 0, 1.

Graph Theory Ch. 1. Fundamental Concept 94 Recursive condition  Similarly, to test , we seek a realization with a vertex w of degree 3 having three neighbors of degree Delete this Vertex A new degree sequence

Graph Theory Ch. 1. Fundamental Concept 95 Recursive condition This exists if and only if is graphic. (See next page) –We reorder this and test –We continue deleting and reordering until we can tell whether the remaining list is realizable. –If it is, then we insert vertices with the desired neighbors to walk back to a realization of the original list. –The realization is not unique.  The next theorem implies that this recursive test work.

Graph Theory Ch. 1. Fundamental Concept 96 Recursive condition

Graph Theory Ch. 1. Fundamental Concept 97 Theorem. For n>1, an integer list d of size n is graphic if and only if d’ is graphic, where d’ is obtained from d by deleting its largest element  and subtracting 1 from its  next largest elements. The only 1-element graphic sequence is d 1 = Proof: 1/6  For n=1, the statement is trivial.  For n>1, we first prove that the condition is sufficient. –Give d with d 1  …..  d n and a simple graph G’ with degree sequence d’ For Example: We have: 1) d= ) G’ with d’ = We show: d is graphic G’

Graph Theory Ch. 1. Fundamental Concept 98 Theorem. For n>1, an integer list d of size n is graphic if and only if d’ is graphic, where d’ is obtained from d by deleting its largest element  and subtracting 1 from its  next largest elements. The only 1-element graphic sequence is d 1 = Proof: 2/6 –We add a new vertex adjacent to vertices in G’ with degrees d 2 -1,…..,d  –These d i are the  largest elements of d after (one copy of)  itself, –Note : d 2 -1,…..,d  need not be the  largest numbers in d’ (see example in previous page) G’ New added vertex d : d 1, d 2, … d n d’ : d 2 -1,…..,d  +1 -1,… d n May not be the  largest numbers

Graph Theory Ch. 1. Fundamental Concept 99 Theorem continue  To prove necessity, 3/6 –Given a simple graph G realizing d, we produce A simple graph G’ realizing d’. –Let w be a vertex of degree  in G, and let S be a set of  vertices in G having the “desired degrees” d 2,…..,d  +1. d : d 1, d 2, …d , d  +1, … d n S:  vertices d1=d1= w

Graph Theory Ch. 1. Fundamental Concept 100 Theorem Proof: continue 4/6 –If N(w)=S, then we delete w to obtain G’. d : d 1, d 2, …d , d  +1, … d n  Vertices, N(w)=S i.e. They are connected to w d1=d1= w Delete w than we have d’ : d 2 -1,…..,d  +1 -1, … d n

Graph Theory Ch. 1. Fundamental Concept 101 Theorem Proof: continue 5/6 –Otherwise, Some vertex of S is missing from N(w). In this case, we modify G to increase |N(w)  S| without changing any vertex degree. Since |N(w)  S| can increase at most  times, repeating this converts G into another graph G* that realizes d and has S as the neighborhood of w. From G* we then delete w to obtain the desired graph G’ realizing d’. d : d 1, d 2, …d , d  +1, … d n  Vertices, N(w)  S i.e. Some vertices are not connected to w. - We make them become connected to w without changing their degree. d1=d1= w

Graph Theory Ch. 1. Fundamental Concept Theorem Proof: continue6/6 To find the modification when N(w)  S, we choose x  S and z  S so that w z are connected and w x are not. We want to add wx and delete wz, but we must preserve vertex degrees. Since d(x)>d(z) and already w is a neighbor of z but not x, there must be a vertex y adjacent to x but not to z. Now we delete {wz,xy} and add {wx,yz} to increase |N(w)  S|. w z x y This y must exist. w z x y  It becomes connected w

Graph Theory Ch. 1. Fundamental Concept switch  A 2-switch is the replacement of a pair of edges xy and zw in a simple graph by the edges yz and wx, given that yz and wx did not appear in the graph originally.

Graph Theory Ch. 1. Fundamental Concept 104 Theorem: If G and H are two simple graphs with vertex set V, then d G (v)=d H (v) for every v  V if and only if there is a sequence of 2-switches that transforms G into H Proof:  Every 2-switch preserves vertex degrees, so the condition is sufficient.  Conversely, when d G (v)=d H (v) for all v  V, we obtain an appropriate sequence of 2-switches by induction on the number of vertices, n.  If n<3, then for each d 1,…..,d n there is at most one simple graph with d(v i )=d i.  Hence we can use n=3 as the basis step.

Graph Theory Ch. 1. Fundamental Concept 105 Theorem (Continue)  Consider n  4, and let w be a vertex of maximum degree, .  Let S={v 1,…..,v  } be a fixed set of vertices with the  highest degrees other than w.  As in the proof of Theorem , some sequence of 2- switches transforms G to a graph G* such that N G* (w)=S, and some such sequence transforms H to a graph H* such that N H* (w)=S.  Since N G* (w)=N H* (w), deleting w leaves simple graphs G’=G*-w and H’=H*-w with d G’ (v)=d H’ (v) for every vertex v.

Graph Theory Ch. 1. Fundamental Concept 106 Theorem Continue  By the induction hypothesis, some sequence of 2-switches transforms G’ to H’. Since these do not involve w, and w has the same neighbors in G* and H*, applying this sequence transforms G* to H*.  Hence we can transform G to H by transforming G to G*, then G* to H*, then (in reverse order) the transformation of H to H*.

Graph Theory Ch. 1. Fundamental Concept 107 Directed Graph and Its edges  A directed graph or digraph G is a triple: –A vertex set V(G), –An edge set E(G), and –A function assigning each edge an ordered pair of vertices. The first vertex of the ordered pair is the tail of the edge The second is the head Together, they are the endpoints.  An edge is said to be from its tail to its head. –The terms “head” and “tail” come from the arrows used to draw digraphs.

Graph Theory Ch. 1. Fundamental Concept 108 Directed Graph and its edges  As with graphs, we –assign each vertex a point in the plane and –each edge a curve joining its endpoints.  When drawing a digraph, we give the curve a direction from the tail to the head.

Graph Theory Ch. 1. Fundamental Concept 109 Directed Graph and its edges  When a digraph models a relation, each ordered pair is the (head, tail) pair for at most one edge. –In this setting as with simple graphs, we ignore the technicality of a function assigning endpoints to edges and simply treat an edge as an ordered pair of vertices.

Graph Theory Ch. 1. Fundamental Concept 110 Loop and multiple edges in directed graph  In a graph, a loop is an edge whose endpoints are equal.  Multiple edges are edges having the same ordered pair of endpoints.  A digraph is simple if each ordered pair is the head and tail of the most one edge; one loop may be present at each vertex. Loop Multiple edges

Graph Theory Ch. 1. Fundamental Concept 111 Loop and multiple edges in directed graph  In the simple digraph, we write uv for an edge with tail u and head v. –If there is an edge form u to v, then v is a successor of u, and u is a predecessor of v. –We write u  v for “there is an edge from u to v”. Predecessor Successor

Graph Theory Ch. 1. Fundamental Concept 112 Path and Cycle in Digraph  A digraph is a path if it is a simple digraph whose vertices can be linearly ordered so that there is an edge with tail u and head v if and only if v immediately follows u in the vertex ordering.  A cycle is defined similarly using an ordering of the vertices on the cycle.

Graph Theory Ch. 1. Fundamental Concept 113 Underlying graph  The underlying graph of a digraph D: –the graph G obtained by treating the edges of D as unordered pairs; –the vertex set and edges set remain the same, and the endpoints of an edge are the same in G as in D, –but in G they become an unordered pair. The underlying GraphA digraph

Graph Theory Ch. 1. Fundamental Concept 114 Underlying graph  Most ideals and methods of graph theorem arise in the study of ordinary graphs.  Digraphs can be a useful additional tool, especially in applications  When comparing a digraph with a graph, we usually use G for the graph and D for the digraph. When discussing a single digraph, we often use G.

Graph Theory Ch. 1. Fundamental Concept 115 Adjacency Matrix and Incidence Matrix of a Digraph  In the adjacency matrix A(G) of a digraph G, the entry in position i, j is the number of edges from v i to v j.  In the incidence matrix M(G) of a loopless digraph G, we set m i,j =+1 if v i is the tail of e j and m i,j = -1 if v i is the head of e j.

Graph Theory Ch. 1. Fundamental Concept 116 Example of adjacency matrix  The underlying graph of the digraph below is the graph of Example ; note the similarities and differences in their matrices.

Graph Theory Ch. 1. Fundamental Concept 117 Connected Digraph  To define connected digraphs, two options come to mind. We could require only that the underlying graph be connected.  However, this does not capture the most useful sense of connection for digraphs.

Graph Theory Ch. 1. Fundamental Concept 118 Weakly and strongly connected digraphs  A graph is weakly connected if its underlying graph is connected.  A digraph is strongly connected or strong if for each ordered pair u,v of vertices, there is a path from u to v.

Graph Theory Ch. 1. Fundamental Concept 119 Eulerian Digraph  An Eulerian trail in digraph (or graph) is a trail containing all edges.  An Eulerian circuit is a closed trail containing all edges.  A digraph is Eulerian if it has an Eulerian circuit.

Graph Theory Ch. 1. Fundamental Concept 120 Lemma. If G is a digraph with  + (G)  1, then G contains a cycle. The same conclusion holds when  - (G)  Proof.  Let P be a maximal path in G, and u be the last vertex of P.  Since P cannot be extended, every successor of u must already be a vertex of P.  Since  + (G)  1, u has a successor v on P.  The edge uv completes a cycle with the portion of P from v to u.

Graph Theory Ch. 1. Fundamental Concept 121 Theorem: A digraph is Eulerian if and only if d + (v)=d - (v) for each vertex v and the underlying graph has at most one nontrivial component

Graph Theory Ch. 1. Fundamental Concept 122 De Bruijn cycles  Application: –There are 2 n binary strings of length n. –Is there a cyclic arrangement of 2 n binary digits such that the 2 n strings of n consecutive digitals are all distinct? –Example: For n=4, ( ) works …

Graph Theory Ch. 1. Fundamental Concept 123 De Bruijn cycles  We can use such an arrangement to keep track of the position of a rotating drum. –One drum has 2 n rotational positions. –A band around the circumference is split into 2 n portions that can be coded 0 or 1. –Sensors read n consecutive portions. –If the coding has the property specified above, then the position of the drum is determined by the string read by the sensors.

Graph Theory Ch. 1. Fundamental Concept 124 De Bruijn cycles  To obtain such a circular arrangement, –define a digraph D n whose vertices are the binary (n-1)-tuples. –Put an edge from a to b if the last n-2 entries of a agree with the first n-2 entries of b. –Label the edge with the last entry of b.  Below we show D 4. We next prove that D n is Eulerian and show how an Eulerian circuit yields the desired circular arrangement.

Graph Theory Ch. 1. Fundamental Concept 125 De Bruijn cycles a b Put an edge from a to b if the last n-2 entries of a agree with the first n-2 entries of b. Label the edge with the last entry of b.

Graph Theory Ch. 1. Fundamental Concept 126 Theorem. The digraph D n of Application is Eulerian, and the edge labels on the edges in any Eulerian circuit of D n form a cyclic arrangement in which the 2 n consecutive segments of length n are distinct Proof:  We show –first that D n is Eulerian. –Then the labels on the edges in any Eulerian circuit of D n form a cyclic arrangement in which the 2 n consecutive segments of length n are distinct.

Graph Theory Ch. 1. Fundamental Concept 127 Theorem. The digraph D n is Eulerian Proof: 1/2  Every vertex has out-degree 2 –because we can append a 0 or a 1 to its name to obtain the name of a successor vertex.  Similarly, every vertex has in-degree 2, –because the same argument applies when moving in reverse and putting a 0 or a 1 on the front of the name

Graph Theory Ch. 1. Fundamental Concept 128 Theorem. The digraph D n is Eulerian Proof: 2/2  Also, D n is strongly connected, –because we can reach the vertex b=(b 1,…..,b n-1 ) from any vertex by successively follows the edges labeled b 1,…..,b n-1.  Thus D n satisfies the hypotheses of Theorem and is Eulerian a b 1 1

Graph Theory Ch. 1. Fundamental Concept 129 Theorem. The labels on the edges in any Eulerian circuit of D n form a cyclic arrangement in which the 2 n consecutive segments of length n are distinct Proof: 1/4  Let C be an Eulerian circuit of D n. Arrival at vertex a=(a 1,…..,a n-1 ) must be along an edge with label a n-1 –because the label on an edge entering a vertex agrees with the last entry of the name of the vertex a b

Graph Theory Ch. 1. Fundamental Concept 130 Theorem. The labels on the edges in any Eulerian circuit of D n form a cyclic arrangement in which the 2 n consecutive segments of length n are distinct Proof: 2/4  The successive earlier labels (looking backward) must have been a n-2,…..,a 1 in order. –because we delete the front and shift the reset to obtain the reset of the name at the head a b

Graph Theory Ch. 1. Fundamental Concept 131 Theorem. The labels on the edges in any Eulerian circuit of D n form a cyclic arrangement in which the 2 n consecutive segments of length n are distinct Proof: 2/4  If C next uses an edge with label a n, then the list consisting of the n most recent edge labels at that time is a 1,…..a n a b 0 1

Graph Theory Ch. 1. Fundamental Concept 132 Theorem. The labels on the edges in any Eulerian circuit of D n form a cyclic arrangement in which the 2 n consecutive segments of length n are distinct Proof: 3/4  Since –the 2 n-1 vertex labels are distinct, and –the two out-going edges have distinct labels, and –we traverse each edge exactly once Distinct vertex label Distinct labels on out-going edges

Graph Theory Ch. 1. Fundamental Concept 133 Theorem. The labels on the edges in any Eulerian circuit of D n form a cyclic arrangement in which the 2 n consecutive segments of length n are distinct Proof: 4/4  We have shown that the 2 n strings of length n in the circular arrangement given by the edge labels along C are distinct.