Graph Theory Judith Stumpp 6. November 2013 Bei dem folgenden Skript handelt es sich um einen Mitschrieb der Vorlesung Graph Theory vom Winter- semester 2011/2012. Sie wurde gehalten von Prof. Maria Axenovich Ph.D. Der Mitschrieb erhebt weder Anspruch auf Vollständigkeit, noch auf Richtigkeit! 1 Kapitel 1 Definitions The graph is a pair V; E. V is a finite set and E V a pair of elements in V . V is called the set of vertices ⊆ 2 and E the set of edges. 1 2 5 Visualize: G = (V; E);V = 1; 2; 3; 4; 5 ;E = 1; 2 ; 1; 3 ; 2; 4 3 f g ff g f g f gg 4 History: word: Sylvester (1814-1897) and Cayley (1821-1895) Euler - developed graph theory Königsberg bridges (today Kaliningrad in Russia): A A D C D C B B Problem: Travel through each bridge once, come back to the original point. Impossible! Notations: V Kn= (V; ) - complete graph on n vertices V = n • 2 j j v1 v2 v6 v3 v5 v4 K5 K3 C6 = v1v2v3v4v5v6 Cn - cycle on n vertices • V = v1; v2; : : : ; vn ;E = v1; v2 ; v2; v3 ;:::; vn−1; vn ; vn; v1 f g ff g f g f g f gg n Pn - path on n vertices (Note: P . path on n edges (Diestel)) • V = v1; v2; : : : ; vn ;E = v1; v2 ; v2; v3 ;:::; vn−1; vn f g ff g f g f gg 2 Let P be a path from v1 to vn. The subpath of P from vi to vj is viP vj and the subpath from vi+1 to • ◦ vj is viP vj. En= (V; ); V = n isolated vertices • ; j j m n E10 Km,n Kn;m= (A B; A B);A B = complete bipartite graph • [ × \ ; Peterson graph: V = f1;2;3;4;5g ;E = i; j ; k; l : i; j k; l = • 2 fff g f g f g \ f g ;g 1, 2 { } 4, 5 3, 5 3, 4 { } { } { } 1, 3 2, 5 { } { } 1, 4 2, 4 { } { } 2, 3 1, 5 { } { } Kneser Graph K(n; k)= ( V ;E) • k V = n; E = A; B : A; B V and A B = : j j ff g 2 k \ ;g V is the set of k-element subsets of V, V = jV j k j k j k Q n- hypercube of dimension n. • f1;2;:::;ng Qn = 2 ;E ;E = A; B : A B = 1 (A B := (A B) (A B)) f g ff g j 4 j g 4 [ − \ V - set of binary n-tuples E - pairs of binary tuples different in 1 position Q1 Q2 Q3 V = ; 1 = (0); (1) V = (0; 0); (0; 1); (1; 0); (1; 1) V = (0; 0; 0);:::; (1; 1; 1) f; f gg f g f g f (1,1,1) g (1,0,1) (1,1) (1,1,0) (0,1,1) 1 (1,0,0) (0,0,1) (0,1) (1,0) (0,1,0) 0 (0,0) (0,0,0) Qn : (1,..., 1) n n 1 weight n-1 − n n 2 1001 0001 n 3 Qn 1 n weight 2 Qn 1 − 2 weight 1 (# 1’s in a binary tuple) − n (0,..., 0) 3 Parameter: Let G = (V; E) be a graph. The order of G ist the number of vertices ( V ) and the size of G j j is the number of edges ( E ). j j If the order of G is n, then 0 size(G) n . ≤ ≤ 2 If e = x; y E, x is adjacent to y and x is incident to e. f g 2 There is a n n matrix A of G = ( v1; : : : ; vn ;E) which is called the adjacent matrix. × f g v1 0 1 1 0 1 0 1 1 For v3 v2 v4 A = 0 1. 1 1 0 0 B0 1 0 0C B C @ A Subgraph: H G; H = (V 0;E0);G = (V; E);V 0 V; E0 E ⊆ ⊆ ⊆ v1 v1 v3 v2 v4 v3 v2 H H is an induced subgraph of G if H G and for v1; v2 V (H): v1; v2 E(H) v1; v2 E(G). ind⊆ ⊆ 2 f g 2 , f g 2 In the upper example it is no induced subgraph. An induced subgraph is obtained from G by deleting vertices. E.g.: v1 v1 v1 v1 v v3 v2 v4 v3 2 v2 v4 v3 v2 v4 v3 v4 Let G = (V; E) and G0 = (V 0;E0) be graphs. Then we define G G0 := (V C0;E E0) and G G0 := [ [ [ \ (V C0;E E0). \ \ G[X] := (X; x; y : x; y X; x; y E(G) ) is called the subgraph of G induced by a vertex set ff g 2 f g 2 g X V (G). E.g.: ⊆ G 2 G[ 1, 2, 3, 4 ] 2 { } 4 1 3 5 4 1 3 A degree d(v) = deg v of a vertex is the number of edges incident to that vertex. v1 v v3 2 v4 deg v1 = 2; deg v2 = 3; deg v3 = 2; deg v4 = 1 In this example the degree sequence is (2; 3; 2; 1), the minimum degree δ(G) is 1 and the maximum degree ∆(G) is 3. n 1 Apparently E(G) = 2 deg vi is true. j j i=1 n P Thus deg vi is even and therefore the number of vertices with odd degree is even. i=1 P n 1 d(G) := n deg vi is called the average degree of G. i=1 P 4 2 Extremal graph theorem: We’ll prove that if G has n vertices and > n edges G has a 4 ) triangle. l m Let A; B V; A B = . P is an A-B-path if P = v1 : : : vk;V (P ) A = v1 and V (P ) B = vk . ⊆ \ ; \ f g \ f g A graph is connected if any two vertices are linked by a path. A maximal connected subgraph of a graph is a connected component. A connected graph without cycles is called a tree. A graph without cycles (acyclic graph) is called a forest. Other „special named“ graphs: star caterpillar spider broom Proposition: If a graph G has a minimum degree δ(G) 2 then G has a path of length δ(G) and a cycle with at least δ(G) + 1 vertices. ≥ proof: Let P = (x0; : : : ; xk) be a longest path in G. Then all neighbors of xk are in V (P ) (y is a neighbor of x if x; y E). In particular k δ(G). f g 2 ≥ Let i = min j 0; : : : ; k : xk; xj E . Then xixkxk−1 : : : xi is a cycle of length at least δ + 1. f 2 f g f g 2 g δ ≥ xk xi x0 The girth of a graph G is the length of a smallest cycle in G. The distance dG(v; w) of v; w G is the length of the smallest path between them (min = ). 2 ; 1 The diameter of G is max dG(v; w): v; w G . f 2 g Proposition: Every nontrivial tree T has a leaf. proof: Assume T has no leaves. T has no isolated vertices δ(T ) 2 Cn T ) ≥ ) ⊆ - A tree T of order n 1 has n 1 edges. ≥ − proof: T = K1 X Assume it holds for all trees of order < n. Let v be a leaf of T , T 0 := T v: T 0 = n 1 < n. − T)0 is j acyclic.j − 0 0 0 Let v ; w T : P v = v0; v1; : : : ; vn = w T . 2 9 ⊆ To show: vi = v for all i = 0; : : : ; n 6 0 0 v0; vn = v because v0; vn T ; v T 6 2 62 vi = v (i = 1; : : : ; n 1) because dT (vi) 2, vi is not a leaf. 6 − ≥ 5 0 0 P T connecting v0 and w T connected T is a tree. ) ⊆ ) ) With induction hypothesis T 0 has (n 1) 1 edges. Thus T has (n 1) 1 + 2 = n 1 edges. − − − − − A walk is an alternating sequence v0e0v1e1 : : : vn of vertices and edges so that ei = vivi+1 for all n = 0; : : : ; n 1. Compared to a path it is allowed to pass edges and vertices more than once. If − v0 = vn, then the walk is a closed walk. - If G has a u-v-walk (between vertices u; v) G has a u-v-path. ) proof: Consider the shortest walk between u and v is W . Then W is a path. If not, W has a repeated 0 vertex W = ue0v1e1 ::: vi ::: vi : : : v, then W = W1W2 is a shorter u-v-walk. =:W1 =:W~ =:W2 - If G has an| odd{z closed} | {zwalk} | {z (i.e.} odd # edges) then G has an odd cycle. proof: If there are no repeated vertices (except for first and last) we have an odd cycle. ) If there is a repeated vertex vi;W = v0e0v1 ::: vi : : : vi : : : vn = v0 : 1’st part of W2 W1 2’nd part of W2 W is a union of two closed walks W1 and W2. Either W1 or W2 is an odd closed walk | {z } | {z } | {z } by induction it contains an odd cycle. ) - If G has a closed walk with a non-repeated edge W = v0e0v1 : : : ei : : : ei is unique, then G contains a cycle. proof: Induction on # vertices. Basis: Step: W = v0e0v1 ::: vi : : : vi : : : vn = v0 1’st part of W2 W1 2’nd part of W2 (note, there is a repeated vertex vj, otherwise W is a cycle) | {z } | {z } | {z } So, W is a union of two closed walks W1 and W2 and either W1 or W2 has a non-repeated edge.