Graph Theory Lecture by Prof. Dr. Maria Axenovich Lecture notes by M´onika Csik´os,Daniel Hoske and Torsten Ueckerdt 1 Contents 1 Preliminaries4 2 Matchings 17 3 Connectivity 25 4 Planar graphs 36 5 Colorings 52 6 Extremal graph theory 64 7 Ramsey theory 75 8 Flows 86 9 Random graphs 93 10 Hamiltonian cycles 99 References 101 Index 102 2 Introduction These notes include major definitions, theorems, and proofs for the graph theory course given by Prof. Maria Axenovich at KIT during the winter term 2019/20. Most of the content is based on the book \Graph Theory" by Reinhard Diestel [4]. A free version of the book is available at http://diestel-graph-theory.com. Conventions: • G = (V; E) is an arbitrary (undirected, simple) graph • n := jV j is its number of vertices • m := jEj is its number of edges Notation notation definition meaning V k , V finite set, fS ⊆ V : jSj = kg the set of all k-element k integer subsets of V V 2, V finite set f(u; v): u; v 2 V; u 6= vg the set of all ordered pairs of elements in V [n], n integer f1; : : : ; ng the set of the first n posi- tive integers N 1; 2;::: the natural numbers, not including 0 2S, S finite set fT : T ⊆ Sg the power set of S, i.e., the set of all subsets of S S4T , S, T finite sets (S [ T ) n (S \ T ) the symmetric difference of sets S and T , i.e., the set of elements that ap- pear in exactly one of S or T A[_ B, A, B disjoint sets A [ B the disjoint union of A and B 3 1 Preliminaries Definition 1.1. A graph G is an ordered pair (V; E), where V is a finite set and graph, G V E ⊆ 2 is a set of pairs of elements in V . • The set V is called the set of vertices and E is called the set of edges of G. vertex, edge V • The edge e = fu; vg 2 2 is also denoted by e = uv. • If e = uv 2 E is an edge of G, then u is called adjacent to v and u is called adjacent, incident incident to e. • If e1 and e2 are two edges of G, then e1 and e2 are called adjacent if e1 \ e2 6= ;, i.e., the two edges are incident to the same vertex in G. We can visualize graphs G = (V; E) using pictures. For each vertex v 2 V we draw a point (or small disc) in the plane. And for each edge uv 2 E we draw a continuous curve starting and ending in the point/disc for u and v, respectively. Several examples of graphs and their corresponding pictures follow: V = [5], E = f12; 13; 24g V = fA; B; C; D; Eg, E = fAB; AC; AD; AE; CEg Definition 1.2 (Graph variants). • A directed graph is a pair G = (V; A) where V is a finite set and A ⊆ V 2. The directed graph edges of a directed graph are also called arcs. arc • A multigraph is a pair G = (V; E) where V is a finite set and E is a multiset of multigraph V V elements from 1 [ 2 , i.e., we also allow loops and multiedges. • A hypergraph is a pair H = (X; E) where X is a finite set and E ⊆ 2X n f;g. hypergraph Definition. For two graphs G1 = (V1;E1) and G2 = (V2;E2) we say that G1 and G2 are isomorphic, denoted by G1 ' G2, if there exists a bijection φ : V1 ! V2 with isomorphic, ' xy 2 E1 if and only if φ(x)φ(y) 2 E2. Loosely speaking, G1 and G2 are isomorphic if they are the same up to renaming of vertices. When making structural comments, we do not normally distinguish between isomor- phic graphs. Hence, we usually write G1 = G2 instead of G1 ' G2 whenever vertices = 4 are indistinguishable. Then we use the informal expression unlabeled graph (or just unlabeled graph graph when it is clear from the context) to mean an isomorphism class of graphs. Important graphs and graph classes Definition. For all natural numbers n we define: • the complete graph Kn on n vertices as the (unlabeled) graph isomorphic to complete graph, [n] Kn [n]; 2 . We also call complete graphs cliques. • for n ≥ 3, the cycle Cn on n vertices as the (unlabeled) graph isomorphic to cycle, Cn [n]; fi; i + 1g : i = 1; : : : ; n − 1 [ n; 1 . The length of a cycle is its number of edges. We write Cn = 12 : : : n1. The cycle of length 3 is also called a triangle. triangle • the path Pn on n vertices as the (unlabeled) graph isomorphic to [n]; fi; i+1g : path, Pn i = 1; : : : ; n − 1 . The vertices 1 and n are called the endpoints or ends of the path. The length of a path is its number of edges. We write Pn = 12 : : : n. • the empty graph En on n vertices as the (unlabeled) graph isomorphic to [n]; ; . empty graph, En Empty graphs correspond to independent sets. 5 • for m ≥ 1, the complete bipartite graph Km;n on n+m vertices as the (unlabeled) complete bipartite graph isomorphic to (A [ B; fxy : x 2 A; y 2 Bg), where jAj = m and jBj = n, graph, Km;n A \ B = ;. • for r ≥ 2, a complete r-partite graph as an (unlabeled) graph isomorphic to complete r-partite A1[···_ [_ Ar; fxy : x 2 Ai; y 2 Aj; i 6= jg ; where A1;:::;Ar are non-empty finite sets. In particular, the complete bipartite graph Km;n is a complete 2-partite graph. • the Petersen graph as the (unlabeled) graph isomorphic to Petersen graph [5] [5] ; fS; T g : S; T 2 ;S \ T = ; : 2 2 • for a natural number k, k ≤ n, the Kneser graph K(n; k) as the (unlabeled) Kneser graph, graph isomorphic to K(n; k) [n] [n] ; fS; T g : S; T 2 ;S \ T = ; : k k Note that K(5; 2) is the Petersen graph. 6 • the n-dimensional hypercube Qn as the (unlabeled) graph isomorphic to hypercube, Qn 2[n]; fS; T g : S; T 2 2[n]; jS4T j = 1 : Vertices are labeled either by corresponding sets or binary indicators vectors. For example the vertex f1; 3; 4g in Q6 is coded by (1; 0; 1; 1; 0; 0; 0). Basic graph parameters and degrees Definition 1.3. Let G = (V; E) be a graph. We define the following parameters of G. • The graph G is non-trivial if it contains at least one edge, i.e., E 6= ;. Equiva- non-trivial lently, G is non-trivial if G is not an empty graph. • The order of G, denoted by jGj, is the number of vertices of G, i.e., jGj = jV j. order, jGj • The size of G, denoted by kGk, is the number of edges of G, i.e., kGk = jEj. size, kGk n Note that if the order of G is n, then the size of G is between 0 and 2 . • Let S ⊆ V be a set of vertices. The neighbourhood of S, denoted by N(S), is the neighbourhood, set of vertices in V that have an adjacent vertex in S. The elements of N(S) are N(v) called neighbours of S. Instead of N(fvg) for v 2 V we usually write N(v). neighbour 7 • If the vertices of G are labeled v1; : : : ; vn, then there is an n × n matrix A with entries in f0; 1g, which is called the adjacency matrix and is defined as follows: adjacency matrix vivj 2 E , A[i; j] = 1 0 1 0 1 1 0 B C B C B1 0 1 1C A = B C B C B1 1 0 0C @ A 0 1 0 0 A graph and its adjacency matrix. • The degree of a vertex v of G, denoted by d(v) or deg(v), is the number of degree, d(v) edges incident to v. deg(v1) = 2, deg(v2) = 3, deg(v3) = 2, deg(v4) = 1 • A vertex of degree 1 in G is called a leaf , and a vertex of degree 0 in G is called leaf an isolated vertex. isolated vertex • The degree sequence of G is the multiset of degrees of vertices of G, e.g. in the degree sequence example above the degree sequence is f1; 2; 2; 3g. • The minimum degree of G, denoted by δ(G), is the smallest vertex degree in G minimum degree, (it is 1 in the example). δ(G) • The maximum degree of G, denoted by ∆(G), is the highest vertex degree in G maximum degree, (it is 3 in the example). ∆(G) • The graph G is called k-regular for a natural number k if all vertices have regular degree k. Graphs that are 3-regular are also called cubic. cubic P • The average degree of G is defined as d(G) = v2V deg(v) =jV j. Clearly, we average degree, have δ(G) ≤ d(G) ≤ ∆(G) with equality if and only if G is k-regular for some k. d(G) 8 Lemma 1 (Handshake Lemma, 1.2.1). For every graph G = (V; E) we have X 2jEj = d(v): v2V Proof. Let X = f(e; x): e 2 E(G); x 2 V (G); x 2 eg. Then X jXj = d(x) v2V (G) and X jXj = 2 = 2jE(G)j: e2E(G) The result follows.