Books. Graph theory. Gábor Galambos. Heidelberg Themes: (fourth edition), G. Chartrand, L. Lesniak: Graphs and Digraphps

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1 Books G. Chartrand, L. Lesniak: Graphs and Digraphps (fourth edition), Chapman and Hall/CRC. 5. ISBN Graph theory Gábor Galambos Heidelberg GraphTheory - Béla, Bollobás: Graph Theory,, An Introductory Course. Springer- Verlag,, Berlin Heidelberg ISBN Nemhauser and Wolsey: Integer and Combinatorial Optimization. John Wiley and sons ISBN X. W. Kocay,, D.L. Kreher: : Graphs, Algorithms,, and Optimization. Chapman and Hall / CRC. 5. ISBN J.A. Bondy and U.S.R. Murty: Graph Theory with Application. North-Holland, 976. GraphTheory - B. Bollobás: Extremal Graph Theory, Academic Press, 978. M.C. Golumbic: Algorithmic Graph Theory and Perfect Graphs. Academic Press, 98. R.L. Graham, B.L. Rotschild and J.H. Spencer: Ramsey Theory. Wiley Interscience Publication, 98. T.B. Boffey: Graph Theory in Operation Research. MacMillan Press, London. 98. L.Lovász and M.D. Plummer: Matching Theory. Akadémiai Kiadó, Budapest, 986. T.R. Jensen and B. Toft: Graph Coloring Problems. Wiley Interscience Publication, 995. W.T. Tutte: Graph Theory. Cambridge University Press,. B. Bollobás: Random Graphs, Cambridge University Press,. GraphTheory - Themes: Fundamentals and elementary definitions The structure of graphs Trees and connectivity Eulerian and Hamiltonian graphs and digraphs Planar graphs Graphs and electrical networks Flows, matchings, factors and decompositions Labeling of graphs Dominations in graphs Extremal graph theory Graph colorings Ramsey theory The probabilistic method in graph theory Random graphs GraphTheory -

2 . Fundamentals and Elementary Results Elementary definitions: veretx,, edge, order, size, adjacency etc Connections between order and size of G. Graph isomorhism. Subgraphs, supergraphs,induced subgraphs. Special graphs: trees, complete graph, bipartite graph anda complement graphs. Operations on graphs. Degree squences. Going-over over a graph: walk, trail, path, circuit, cycle. Measurments in a graph: distance, eccentricity, radius, diameter, center. The Shortest Path Problem Connections between measurments. Digraphs and Multigraphs. GraphTheory Elementary definitions A graph G is a finite nonempty set of objects (vertices) together with a (possibly( empty) set of unordered pairs of distinct vertices of G called edges. V(G) v v the vertex set of G e 5 e e e E(G) the edge set of G e = {v, } joins the vertices v and, v and are adjacent vertices, v e 5 and e are adjecent edges, e and v 5 are incident. e We can use the notation v instead of {v{, }. GraphTheory - 6 The cardinality of the vertex set of a graph G is the order of G (n(g)).). e The cardinality of the edge set of a graph G is the size of G (m(g)).). v A graph G with order n and size m is denoted by G(n,m) or G n,m. e e e A graph G with a vertex set V(G) = {v,,, v n } and edge set E(G) = {e, e,, e m } can also be described by means of matrices. The adjacency matrix A(G) = [a ij ij = { if v i v j if v i v j a ij E(G) E(G) The icidence matrix B(G) = [b ij b ij ij ] nxn ij ] nxm ij = { if v i and e j incident otherwise. where where GraphTheory - 7 A = v e 5 v v e e e e e 5 v v A graph and its adjacency and incidence matrices GraphTheory - 8 B = v v

3 Vertex-degree The degree of a vertex v in a graph G is the number of edges of incident with v.. It is denoted by deg G v or deg v. The set of adjacent vertices to v is denoted by The vertex is called even or odd according to whether its degree is even or odd. A vertex is isolated if its degree is, while a vertex is end-vertex if its degree is. (G) = min deg G v is the minimum degree of G. v G (G) = max deg G v is the maximum degree of G. v G (v). deg G v = (v). GraphTheory - 9 v v v 6 v 7 v 8 5 v 5 deg G = deg G = deg G v 9 = (G) = deg G v 8 = GraphTheory - v 9 Vertex-degrees in a graph (G) = deg G v 6 = 5 Theorem.. ( Handshaking Lemma): Let G be a graph of order n and size m where V(G) ) = {v{,,, v n }. Then n i= deg G v i = m Proof: The statement of the theorem follows immediately from the fact that every edge is incident with two vertices. Corollary.. : In any graph, there is an even number of odd vertices. Proof: Let W (U) be the set of odd (even)) vertices. Then deg G v i = deg G v i + deg G v i = m v V(G) v even GraphTheory - even. GraphTheory - W must be also even.. Graph isomorphism A graph G is isomorphic to a graph G if there exists a one-to to-oneone mapping, called isomorphism, from V(G ) onto V(G ) such that preserves adjacency and nonadjacency; ; that is, uv E(G ) iff u v E(GE ). It will be denoted by writing G = G. If G = G then G and G have the same order G and G have the same size every vertex v in G and its image vertex v in G have the same degree in their respective graph The above conditions are necessary but not sufficient for G and G to be isomorphic:

4 v v 5 v v 6 G G G =G. Consider the following mapping v =v, =v, v 6 v 5 v =v 5, v =, Are these graphs isomorphic or not? GraphTheory - v v 6 v 5 v G : V(G ) V(G ): v 5 =, G contains three pairwise adjacent vertices while G does not. So G G. v v 6 =v 6 A graph with no edges is called an empty graph. A graph H is a subgraph of a graph G if V(H) V(G) and E(H) E(G) E(G) We also say that G is a supergraph of H. We delete a vertex v of G if v V(G) and V(G) V(G),, and the subgraph H = G v has vertex set V(H) = V(G) {v}} and E(H) contains all of those edges of G which are not incident with v. If e E(G),, then H = G e (delete an edge) ) is the subgraph of G having vertex set V(H) = V(G) and edge set E(H) = E(G) {e}. GraphTheory - v e G G v G e Whenever a subgraph H of a graph G has the same order as G,, then H is called a spanning subgraph of G. GraphTheory - 5. Induced subgraphs If U is a nonempty subset of the vertex set V(G) of a graph G,, then the subgraph U of G induced by U is the graph having vertex set U and whose edge set consists of those edges of G incident with two elements of U. A subgraph H of G is vertex-induced if H = U for some subset U of V(G). If X is a nonempty subset of E(G),, then the subgraph X induced by X is the graph having edge set X and whose vertex set consists of those vertices of G incident at least one edge of X. A subgraph H of G is edge-induced if H = X for some subset X of E(G). GraphTheory - 6

5 .. Special graphs with special notations A graph G is regular of degree r if deg v = r for each vertex v of G. Such graphs are called r-regular. A graph is complete if every two of its vertices are adjacent. A complete graph of order n and size m is therefore a regular graph of degree n having m = n(n-)/ edges. We denote this graph by K n. The regular graphs of order. GraphTheory - 7 GraphTheory - The Petersen graph 8 Theorem (D. König,, 96): if G is a graph with (G)=d,, then there exists a d-regular graph H containing G as an induced subgraph. G: G : We will prove a stronger result: Theorem.. : for every graph G and every integer r (G), there exists an r-regularregular graph containing G as an induced subgraph. Proof. If G is r-regular,, then there is nothing to prove. So we can suppose that G is not r-regular. Let G be another copy of G and join corresponding vertices whose degrees are less than r.. We call the resulting graph G. If G is r-regularregular then it has the desired properties. If not, we continue this procedure until arriving at an r-regularregular graph G k where k = r (G). GraphTheory - 9 G : A -regular graph containing G as an induced subgraph GraphTheory - 5

6 Is the so constructed r-regularregular graph one of the smallest order with the desired property? Not. Erds and Kelly(96) produced a method for determining the minimum order of r-regularregular graph H containing a given graph G as an induced subgraph. GraphTheory - A complement G of a graph G is that graph such that V( G )= = V(G) V u V(G), uv E( G ) iff uv E(G). Fact : : If G is a graph of order n and size m,, then G is a graph of order n and size n m = m Fact : : The complement ( K n ) of the complete graph K n is the empty graph of order n. A graph is self-complement if G = G. Theorem..: : If G is self-complement complement,, then either n(mod ) or n(mod ). Hint: the size of G is n(n-)/. GraphTheory - A graph G is k-partite, k,, if it is possible to partition V(G) into k subsets V,V,,V,V k (called partite sets) such that every eleme ment of E(G) joins a vertex of V i to a vertex of V j, i j. v v 5 v 7 For k =, the graph is called as bipartite graph. If G is an r-regularregular bipartite graph, r, then V = V. A complete k-partite graph with partite sets V,V, V k, and V i = n i is k-partite graph having the added property that if u V i and v V j then uv E(G).. It is denoted by K(n,n, n k ) or. K n,n,..., nk v v 6 v v v 5 v 7 v 6 V V The complete bipartite graph with V = r and V = s is denoted by K(r,s) or K r,s. (K,s is a star). A bipartite graph with two different (isomorph) representations GraphTheory - GraphTheory - 6

7 .5. Combining graphs to produce new graphs The union of two graphs G = G G has V(G) = V(G ) V(G ) and E(G) = E(G ) E(G ).. (If a graph consists of k disjoint copies of a graph H, then we write G=kH). Example: K K K,. The join of two graphs G = G G has V(G) = V(G ) V(G ) and E(G) = E(G ) V(G ) {uv u V(G ) and v Example: V(G )}. G G G +G GraphTheory - 5 GraphTheory - 6 The Cartesian product of two graphs G = G G has V(G) = V(G ) V(G ) and two vertices (u(,u ) and (v(, ) of G are adjacent iff either u = v and u E(G ) or u = and u v E(G ) Example : u v u (u,u ) (u, ) (v, ) Example : An important class of graphs can be defined in terms of Cartesian products: the n-cube Q n is the graph K if n =. if n then Q n = Q n- K. w w G G G = G G Q Q Q GraphTheory - 7 GraphTheory - 8 7

8 Exercises.. (G. Chartrand and L. Lesniak page -.).). Determine all nonisomorphic graphs of order 5.. Let n be a given positive integer, and let r and s be nonnegative integers such that r+s=n and s is even. Show that there exists a graph G of order n having r even vertices and s odd vertices..6. Degree sequences A sequence d,d, d n if nonnegative integers is called a degree sequence of a graph G if the vertices of can be labelled v,, n so that deg v i = d i for all i. Example:. We saw regular nonisomorphic graphs of order 6 and size 9 at the Lecture. Give another example of two nonisomorhic regular graphs of the same order and same size. A degree sequence:. A nontrivial graph G is called irregular if no two vertices of G have the same degree. Prove that no graph is irregular.,,,, 5. For each integer k, give an example of k nonisomorphic regular graphs, all of the same order and size. GraphTheory - 9 v v v 5 Any permutation of a degree sequence is also degree sequence. GraphTheory - The opposite case: : if a sequence s: d,d,,d n of nonnegative integ- ers are given,, then under what conditions is s a degree sequence of some graphs? If such graph exists then s is called a graphical sequence. Necessary conditions are easy: d i n n- for all i. d i is even. It is easy to see that the sequence,,, is not graphical. GraphTheory - Theorem.5.(Havel, 955, Hakimi,, 96): A sequence s: d,d,,d n of nonnegative integers with d d d n, n, d,, is graphical iff the sequence s : d -,d -,,d d +-,d,d d +,,d n is graphical. Proof: Let us suppose that s is graphical. There exists a graph G of order n- such that s is a degree sequence of G. The vertices of G can be labeled as, n so that d i degvi = GraphTheory - d d + i n. i i d + 8

9 deg = d - Conversely, let s be a graphical sequence. deg v = d - deg = d - v There exist graphs of order n with degree sequence s. deg v d + = d d deg v d + = d d + deg v n = d n G v d +... v d +... v n v deg v = d So the sequence s is graphical Let G be such a graph in which the sum of the degrees of the vertices adjacent to v is maximum. We will show v is adjacent to vertices having degrees d, d,, d d +. deleting v from G we get that the graph G v is graphical. GraphTheory - GraphTheory - Suppose the contrary,, that v is not adjacent to vertices having degrees d, d,, d d +. Graphically: G There exist vertices v r and v s with d r > d s such that v is adjacent to v s but not to v r. v v r Since the degree of v r exceeds that of v s, there exists a vertex v t such that v t is adjacent to v r but not to v s. v v s v t Removing the edges v v s and v r v t and adding the edges v v r and v s v t results in a graph G' ' having the same degree sequence as G. In G' the sum of the degrees of the vertices adjacent to v is larger than that in G.. This is a contradiction. GraphTheory - 5 deg v r = d r > d s = deg v s v d + GraphTheory - 6 9

10 The theorem provides an algorithm for determining whether a given finite sequence of nonnegative integers is graphical. If, upon repeated application of the theorem,, we arrive at a sequence every term of which is,, then the original sequence is graphical. Example: s: 5,,,,,,,,,,. s :,,,,,,,,,. s :,,,,,,,,,. s :,,,,,,,,. s :,,,,,,,. s :,,,,,,. Apply the theorem. Reordering the sequence. s, :,,,,,,,,. s, :,,,,,,,. :,,,,,,. s, s, 5 :,,,,,. s 5 :,,,,,. s 6 :,,,,. s 7 :,,,. s, 6 :,,,,. GraphTheory - 7 GraphTheory - 8 How can we construct a graph G belonging to the sequence s? Theorem.6. (Erd( s and Gallai,, 96): A sequence s: d,d,,d n of nonnegative integers with d d d n, n, d,, is graphical if and only if n = d i i is even and G belonging to s G belonging to s for each integer k, k i= d i k( k ) + n k i= k + min n- { k, d } i G belonging to s GraphTheory - 9 G belonging to s GraphTheory -

11 Exercises.. (G. Chartrand and L. Lesniak page 5.). Determine whether er the following sequences are graphical.. If so, construct a graph with the appropriate degree sequence. a),,,, b),,,,,,, c) 7,7,6,5,,,, d) 7,6,6,5,,,, e) 7,,,,,,,,,. Show that the sequence d,d,,d n is graphical iff the sequence n-d -, n-dn -,, n-dn n - is graphical. GraphTheory -.7. Connected graphs and distance Let u and v be (not neccesseraly distinct) vertices of a graph G. A u v walk W of G is a finite alternating sequence W: u = u,e,u,e,u,.,u k-,e k,u k = v of vertices and edges, beginning with vertex u and ending with vertex v, such that e i = u i- u i for i=,,,k. The number k is called the length of W. u e u e 6 u 6 =v e u e 5 =u 5 u e e GraphTheory - u k=6 Often only the vertices of a walk are indicated since the edges present are then evident. Two u v walks W : u = u,u,u,,u k = v and W :u=v,, = v are equal iff k= and u i =v i for i k. Otherwise W and W are different. Example: A u v walk is closed or open depending on whether u = v or u v. A u-v trail is a u v walk in which no edge is repeated, while a u v path is a u v walk in which no vertex is repeated. Consequence: : every path is a trail. W : v T : v P : v v 5 v 5, v 5, v, v v is a v - walk but that is not a trail. is a trail but is not a path. is a path. GraphTheory - GraphTheory -

12 Theorem.7.: Every u v walk in a graph contains a u v path. Proof. Let W be a u v walk in a graph G. If no vertex of G occurs in W more than once,, then W is a path. Otherwise: u u i+ u j- u u i =u j u k =v GraphTheory - 5 u j G j > n Theorem.8.: : If A is the adjacency matrix of G with V(G) = {v,, n }, then the (i,j( i,j) entry of A k, k, k, is the number of different v i -v j walks of length k in G. Proof: induction on k. If k = then the result is obvious since there exists a v i of length iff v i v j E(G). [ ] ( k ) a v j walk ( k ) Let ijk ( a ) ij = a and assume that ij is the number of differ- ent v i v j walks of length k- in G.. We have a k ij = n t = a ( k ) it tj Every v i v j walk of length k consists of a v i v t walk of length k-, where v t is adjacent to v j. We count all the edges v t v j. By the inductive hypotesis we have the desired result. GraphTheory - 6 a Example: v v A = A = A non-trival closed trail is referred to as a circuit,, and a circuit v,, n (n ) whose vertices v i are distincts called as cycle. An acyclic graph has no cycle. A cycle is even if its length is even; otherwise it is odd. A = W : W : W : W : v v v v A cycle of length n is an n-cycle; ; a -cycle is also called triangle. A vertex u is said to be connected to a vertex v in a graph G if there exists a u v path in G. A graph is connected if every two of its vertices are connected. Otherwise the graph is disconnected. GraphTheory - 7 GraphTheory - 8

13 Let S be a set.. An equivalence relation is a relation between certain pairs of elements of S, which satisfyies the following conditions: Reflexive: x ~ x for all x S. Transitive: : If x ~ y and y ~ z then x ~ z. Symmetric: : If x ~ y then y ~ x. Theorem.9.: : The relation connected to is an equivalence relation on the vertex set of every graph G. Proof.: Homework Each subgraph induced by the vertices in a resulting equivalence class is called a connected component of G.. The number of components of G is denoted by k(g). In a connected graph G the distance d(u) between two vertices u and v is the minimum of the lengths of the u v paths of G. A u v path of length d(u) is called a u v geodesic. GraphTheory - 9 GraphTheory - 5 Under this distance function the set V(G) ) is a metric space, i.e.. the following properties hold: Non-negativity negativity: d(u) d(u) ) = iff u = v. v for all pairs, u of vertices of G,, and Symmetric property: d(u) ) = d(v,u) for all pairs u of vertice of G. Triangle inequality: d(u) ) + d(v,w) d(u,w) for all triples u,w of vertices of G. v v 6 v 7 v 8 v 5 v 9 v v v 6 v v 5 v 9 v 7 v 8 level level level level The distance levels from the vertex v. GraphTheory - 5 GraphTheory - 5

14 .8. The Shortest Path Problem Beside the above defined measurements we can introduce a more practical index: we order to each (u ) E(G) a function w(u) and we call it as weight. The graph which edges have weights is a weighted graph. Let w: E(G) Let H G then a function. We extend this function to subgraphs. w( H ) =e e( H ) w( e ). Many optimization problems amount to finding in a weighted graph a subgraph a certain type with minimum (or maximum) weight. The Shortest Path Problem: given a weighted graph (railway net- work connecting various towns), determine a shortest path (a shortest route) between two specified vertices (town in the network). Let w: E(G) we get: a function. Using the subgraph weight definition, w( P ) =e e( P ) w( e ). Then the distance of two vertices in a weighted graph G is dg,w( u ) = min w( P ), where the summation is over all P(u) in G. We shall refer to the weight of a path in a weighted graph as its length. P GraphTheory - 5 GraphTheory - 5 The Dijkstra algorithm (959) Suppose we want to decide the distance between two vertices, u and v. The algorithm uses a permanently (stepwise) increasing set S i, where i n, and {u } S i V(G) and in each step we order a label to the vertices: l: V(G) {}} so that for v S the label l(v) will give the distance of the vertex v from the vertex u within the induced subgraph S. Initial step: i =, S = { u }, Iteration step: If S i = V(G) then the algorithm terminates., if v= u (v) = w(u ), if u v and ( u ) E(G) otherwise l If S i V(G) ) then ui+ = { vi vi Si, l( vi ) = minv where ties t are broken arbitrary. l( v )} If l(u i+ ) == or u i+ = v If l (u i+ ) < then S i+ = S i {u i+ } and let S i l( z ) = min{ l( z ), l(ui + ) + w(ui +,z ) (ui+,z ) E(G ), z Si+ } GraphTheory - 55 i=i+. GraphTheory - 56

15 Lemma..: If v S i then l(v) is the length of the shortest u v path. Furthermore, among the shortest paths there exists a P'(u ) for which if xy E(P'),, then both x S i and y Proof: induction on i. For i = he statement is true. Suppose that the statement is true for a given i < n.. We need to prove our statement for u i+, since the label of the other vertices in S i do not change any more. The first part of the statement follows immediately from the construction. To prove the second part let us suppose that P is a shortest u u i+ path. We can divide P into two parts: P = P P, where P contains only edges of S i, it terminates in x,, and P has only one outgoing edge (x,z( x,z). GraphTheory - 57 S i Because of the induction hypothesis the weight of the edge (x,z) is w(x,z) l(u i+ ). But z P(u,u i+ ), so z = u i+. Therefore each vertex of P is in S i+. Lemma..: u i+ has the property that its label is the length of the that of shortest u u i+ path whose vertices except u i+ are in S i. Proof. We consider paths in the form: P = u,, x,z,, where u,, x S i and z is an endvertex of P.. (z( is arbitrary.) Because of the Lemma.. among the u x paths there is always a minimal length path for which if e E(P),, then both incident vertices of e are in the actual S i. The minimal length of P has changed when z was chosen as a new element of S i. It has been denoted by u i+. GraphTheory - 58 So, when we relabeled u i+, then the label of x was the length of a u x shortest path. When we have certain x-s to reach u i+, then our algorithm chooses the length of the smallest as a label for u i+. Consequence.5.: At the end of the Dijkstra's s Algorithm the values of the labels are the shortest distances from the starting vertex. Proof. It is clear that the algorithm terminates in at most n steps. If S = V(G) then we can use our lemmas, and so the algorithm is correct. If the labels of each v S i are equal to then there is no edge from S to S. If there is an edge xy then the label of y would changed to finite when x was chosen for S. GraphTheory - 59 What is the complexity of the Dijkstra's s Algorithm? It is polynomial in n: O(n ). Example: u u v 7 v v = { v v S, l( v ) i+ i i i i = Step l(u ) l(v )l(vl( )l(v ) l(v ) 7 7 min v S i l( v )} GraphTheory - 6 u u S i 7 u v 6 u v v 6 u v v v l( z ) = min{ l( z ), l(ui + ) + w(ui +,z ) (ui+,z ) E(G ), z Si+ } 5

16 Theorem.6.: : A nontrivial graph is bipartite iff it contains no odd cycles. Proof. A: Suppose that G is bipartite.. Let V and V the two vertex classes.. Let v,, t be a cycle. Let v V. Then V, v V,. Generally, v i V iff i is odd.. Since v t must be in V, so t is even. GraphTheory - 6 The eccentricity e(v) of a vertex v is max u V(G) d(u). The radius (rad G) is the minimum eccentricity among the vertices of G, while the diameter of G (diam G) ) is the maximum eccentricity. B: Suppose that G does not contain odd cycle. Let u V(G) an arbitrary vertex. Let V := {v:{ d(u) is odd} } and V := V V There is no edge between two vertices in the same vertex- class.. (If it is then there is an odd cycle in G.) So,, the graph is bipartite. Theorem.7.: A graph G has radius iff G contains a vertex adjac- ent to all other vertices of G. A vertex v is a central vertex if e(v) ) = rad G and the center Cen(G) is the subgraph of G induced by ist central vertices. GraphTheory - 6 V(G Theorem.8.: For every connected graph G, rad G diam G rad G. G Proof. The left hand side is trivial. To prove the right hand side we choose u and v such that d(u) ) = diam G.. Let w be a central vertex of G. d(u) d(u,w) ) + d(w) e(w) = rad G. How good is this result? How sharp is the upper bound? If diam G < rad G for every graph G then the upper bound may not be sharp. If diam G = rad G for every graph G then the upper bound may be sharp. There are certain levels of sharpness sharpness : There are graphs G for which diam G < rad G and graphs H for which diam H = rad H. There exists an infinite class then diam H = rad H. of graphs H such that if H Such a class exists. For example let type K +. t K consist of the graphs of the t= t= K + K K + K For all of these type of graphs diam G = and rad G =. Is there such a class for which these parameters (diameter and radius) ) are not constants? GraphTheory - 6 GraphTheory - 6 6

17 A more satisfactory class is the class of paths P k+, k,, where diam P k+ = k and rad P k+ = k. k= P 5 What is the situation with the center? Let us consider the following graph: u w diam P k+ = and rad P k+ =. v Cen(G)=K GraphTheory - 65 It is easy to see that Cen(P k+ )=K and Cen(P k )=K. This shows us that there are many graphs that are centers of graphs. GraphTheory - 66 How many? Theorem.9.: Every graph is the center of some connected graph. Proof. Let G be a given graph. We construct a graph H from G by adding four new vertices u, v, u, and for i =,, every vertex of G is joined to v i, and u i is joined to v i. Since e H (u i ) = and e H (v i ) = for i =,, while e H (x) ) = for every vertex x of G,, it follows that Cen(H)= G. G u v u GraphTheory - 67 Exercises.. (G. Chartrand and L. Lesniak page -5.) 5.). Let u and v be arbitrary vertices of a connected graph G. Show that there exists a u v walk containing all vertices of G.. Let G be a graph of order n such that deg v (n-)/ for every v V(G). Prove that G is connected. Examine the shapness of the bound!. Prove that a graph is connected iff for every partition V(G)=V V, there exists an edge of G joining a vertex of V and a vertex of V.. Prove that if G is agraph with (G), then G contain a cycle. 5. Prove that every graph G has a path of length (G). GraphTheory

18 6. Let G be a nontrival connected graph that is not bipartite. Show that G contains adjacent vertices u and v such that deg u + deg v is even. 7. Show that if G a graph of order n and size n /, then either G conta-ins an odd cycle or = K. G n /,n / 8. Prove that if G is a disconnected graph, then G is connected and, in fact, diam G. 9. If k is an integer such that rad G k there is a vertex w such that e(w) = k. diam G, then show that.9. Digraphs and Multigraphs A directed graph or digraph D is a finite nonempty set of objects called vertices together with a set of ordered pairs of distinct vertices of D called arcs or directed edges. The vertex set of D is denoted by V(D) and the arc set is denoted by E(D). Example: V(D)={ )={v,, } and E(D) ) ={(v, ),(v ),(, v ),(v, ),(, )} v v GraphTheory - 69 GraphTheory - 7 Let a = (u( u) be an arc of a digraph D.. Then a is said to join u and v, or we say that a is incident from u and incident to v, while u is incident to a and v is incident from a. Moreover,, u is said to be adjacent to v and adjacent from u. Two vertices u and v are nonadjacent in D,, if u is neither adjacent to nor adjacent from v in D. The outdegree (od v) ) of a vertex v in D is the number of vertices of D that are adjacent from v. The indegree (id v) ) of a vertex v in D is the number of vertices of D that are adjacent to v. The degree (deg v) of a vertex of D is defined by deg v = id v + od v GraphTheory - 7 Theorem..: : If D is a digraph of order n and size m with V(D)= {v,,, n } then n n od v = id v = m. i= i= Proof is trivial. A digraph D is isomorphic to a digraph D if there exists a one-to to- one mapping, called an isomorphism, from V(D ) onto V(D ) such that (u) E(D ) iff ( u, v) E(D ). We denote it by D = D. The relation isomorphic to is an equivalence relation on digraphs. Two digraphs are nonisomorphic,, if they belong to different equivalence classes. There is only one digraph of order (up to isomorphism), this is the trivial digraph.. Also, there is only one digraph of order and size m for each m with m. GraphTheory - 7 8

19 w w A labeled digraph D is a subdigraph of a labeled digraph D if V(D ) V(D) and E(D ) E(D). u w v u w v A subdigraph D of D is a spanning subdigraph if D has the same order as D. A digraph is called symmetric,, if whenever (u)) is an arc of D,, then (v,u)) is also. u v u v A digraph D is called asymmetric digraph or an oriented graph if whenever (u) is an arc of D,, then (v,u) is not an arc of D.. (So, an oriented graph can be obtained from a graph G by assigning a direction to each edge of G.) The digraphs of order and size. GraphTheory - 7 GraphTheory - 7 A digraph D is complete if for every two distinct vertices u and v of D,, at least one of the arcs (u)) and (v,u( v,u) ) is present in D. The complete symmetric digraph of order n has both arcs (u)) and (v,u)) for every two distinct vertices u and v and is denoted by. The underlying graph of a digraph D is that graph obtained by replacing each arc (u) or symmetric pairs (u)) and (v,u( v,u) ) of arcs by the edge of uv. A complete asymmetric digraph is called a tournament. A digraph D is called r-regularregular if od v = id v = r for every vertex v of D. K n A digraph D is connected (or weakly connected) ) if the underlying graph of D is connected. A digraph D is strong (or strongly) connected if for every pair u, v of vertices, D contains both a u v path and a v u path. For vertices u and v in a digraph D containing a u v path, the (directed)) distance d(u) from u to v is the length of a shortest u v path in D. The distances d(u) and d(v,u) are defined for all pairs u, v in a strong digraph. Generally,, the distance is not metric. The distance satisfies the triangle inequality, but it is not symmmetric unless D is symmetric. GraphTheory - 75 GraphTheory

20 A symetric digraph An asymetric digraph If one allows more than one edge (but yet finite number) between the same pair of vertices in a graph, the resulting structure is a multigraph. The edges between the same pair of vertices are called parallel edges If more than one arc in the same direction is permitted to join two vertices in a digraph, a multidigraph results. A loop is an edge (or arc) that joins a vertex to itself. A D regular digraph K complete symetric digraph GraphTheory - 77 GraphTheory - 78 Exercises.. (G. Chartrand and L. Lesniak page -.).). Prove that is connected to is an equivalence relation on the vertex set of a graph.. Prove that is isomorphic to is an equivalence relation on the vertex set of a graph.. Determine all (pairwise( nonisomorphic) digraphs of order and size.. Prove or disprove: : For every integer n,, there exists a digraph D of order n such that for every two distinct vertices of od u od v and id u id v. 5. Prove or disprove: : No digraph contains an odd number of vertices of odd outdegree or an odd number of vertices of odd indegree. GraphTheory - 79