Lecture NotesGraph Theory

Prof. Dr. Maria Axenovich

February 24, 2014

1

ContentsIntroductionIntroduction 3

NotationsNotations 3

1 Preliminaries1 Preliminaries 4

2 Matchings2 Matchings 12

3 Connectivity3 Connectivity 15

4 Planar graphs4 Planar graphs 19

5 Colorings5 Colorings 24

6 Extremal graph theory6 Extremal graph theory 26

7 Ramsey theory7 Ramsey theory 30

8 Flows8 Flows 33

9 Random graphs9 Random graphs 35

10 Hamiltonian cycles10 Hamiltonian cycles 37

LiteratureLiterature 38

Named theoremsNamed theorems 39

IndexIndex 40

2

Introduction

These brief notes include major definitions and theorems of the graph theory lectureheld by Prof. Maria Axenovich at KIT in the winter term 2013/14. We neither provenor motivate the results and definitions. You can look up the proofs of the theoremsin the book “Graph Theory” by Reinhard Diestel [44]. A free version of the book isavailable at http://diestel-graph-theory.comhttp://diestel-graph-theory.com.

Conventions:

• G = (V, E) is an arbitrary (undirected, simple) graph

• n := |V| is its number of vertices

• m := |E| is its number of edges

Notations

notation definition meaning

(Vk ), V finite set,

k integer{S ⊆ V : |S| = k} the set of all k-element

subsets of V

[n], n integer {1, . . . , n} the set of the first n posi-tive integers

2S, S finite set {T : T ⊆ S} the power set of S, i.e. theset of all subsets of S

S△T, S, T finite sets (S ∪ T) \ (S ∩ T) the symmetric differenceof sets S and T, i.e. the setof elements that appear inexactly one of S or T

A ⊔ B A ∪ B the union of the disjointsets A and B

3

1. preliminaries

1 Preliminaries

Definition. A graph, Ggraph G is an ordered pair (V, E), where V is a finite set andE ⊆ (V

2) is a set of pairs of elements in V.

• The set V is called the set of vertex, edgevertices and E is called the set of edges of G.

• The edge e = {u, v} ∈ (V2) is also denoted by e = uv.

• If e = uv ∈ E is an edge of G, then u is called adjacent,incident

adjacent to v and u is calledincident to e.

We can visualize graphs G = (V, E) using pictures. For each vertex v ∈ V we draw apoint (or small disc) in the plane. And for each edge uv ∈ E we draw a continuouscurve starting and ending in the point/disc for u and v, respectively.

Several examples of graphs and their corresponding picture follow:

V = [5], E = {12, 13, 24}

1 2

3

45

V = {A, B, C, D, E},E = {AB, AC, AD, AE, CE}

A

BC D

E

Definition (Graph variants).

• A directed graph directed graphis G = (V, A) where V is a finite set and E ⊆ V2. Theedges of a directed graph are also called arcs arc.

• A multigraph multigraphis G = (V, E) where V is a finite set and E is a multiset ofelements from (V

1) ∪ (V2), i.e. we also allow loops and multiedges.

• A hypergraph hypergraphis H = (X, E) where X is a finite set and E ⊆ 2X \ {∅}.

Definition. For two graphs G1 = (V1, E1) and G2 = (V2, E2) we say that G1 andG2 are isomorphic isomorphic, ≃, denoted by G1 ≃ G2, if there exists a bijection ϕ : V1 → V2 withxy ∈ E1 if and only if ϕ(x)ϕ(y) ∈ E2.

Loosely speaking, G1 and G2 are isomorphic if they are the same up to renaming ofvertices. Hence, we may write G1 = G2 =instead of G1 ≃ G2 whenever vertices areindistinguishable.

4

1. preliminaries

Important graphs and graph classes

Definition. For all natural numbers n we define:

• the complete graph complete graphon n vertices as Kn =([n], ([n]2 )

). Complete graphs are also

called cliques clique.

K5 K3

• for n ≥ 3, the cycle cycleon n vertices as Cn =([n],

{{i, i + 1} : i = 1, . . . , n − 1

}∪{

n, 1})

. The length of a cycle is its number of edges.v1 v2

v3

v4v5

v6

C6 = v1v2v3v4v5v6v1

• the path pathon n vertices as Pn =([n],

{{i, i + 1} : i = 1, . . . , n − 1

}). The vertices

1 and n are called the endpoints or ends of the path. The length of a path is itsnumber of edges.

1 2 3 4 5 6

• the empty graph empty graphon n vertices as En =([n], ∅

). Empty graphs are also called

independent sets independent set.

E10

• for m ≥ 1, the complete bipartite graph completebipartite graph

on n and m vertices as Km,n = (A ∪ B, {xy :x ∈ A, y ∈ B}), where |A| = m and |B| = n, A ∩ B = ∅.

m

nKm,n

5

1. preliminaries

• the Petersen graph Petersen graphas(([5]2 ),

{{S, T} : S, T ∈ ([5]2 ), S ∩ T = ∅

}).

{1, 2}

{3, 4}{3, 5}{4, 5}

{1, 3}

{2, 3}

{1, 4} {2, 4}

{1, 5}

{2, 5}

• for a natural number k, k ≤ n, the Kneser graph Kneser graphas

K(n, k) =((

[n]k

),{{S, T} : S, T ∈

([n]k

), S ∩ T = ∅

}).

Note that K(5, 2) is the Petersen graph.

• the n-dimensional hypercube hypercubeas Qn =(2[n],

{{S, T} : S, T ∈ 2[n], |S△T| = 1

}).

Q1 Q2 Q3

1

0

(1,1)

(1,0)(0,1)

(0,0)

(1,1,1)

(1,1,0) (1,0,1) (0,1,1)

(1,0,0)(0,1,0)

(0,0,1)

(0,0,0)

(1, . . . , 1)

(0, . . . , 0)

weight 1 (# 1’s in a binary tuple)weight 2

weight n-1

n(n

2)

(n3)

(nn2)

( nn−1)

Qn−1Qn−1

00011001

Qn:

6

1. preliminaries

Basic graph parameters and degrees

Definition. Let G = (V, E) be a graph. We define the following parameters ofG.

• The order of G order, |G|, denoted by |G|, is the number of vertices of G, i.e. |G| = |V|.• The size of G size, ∥G∥, denoted by ∥G∥, is the number of edges of G, i.e. ∥G∥ = |E|.

Note that if the order of G is n, then the size of G is between 0 and (n2).

• Let S ⊆ V. The neighbours of S neighbours,N(v)

, denoted by N(S), are the vertices in Vthat have an adjacent vertex in S. Instead of N({v}) for v ∈ V we usuallywrite N(v).

• If the vertices of G are labeled v1, . . . , vn, then there is an n × n matrix Awith entries in {0, 1}, which is called the adjacency matrix adjacency matrixand is defined asfollows:

vivj ∈ E ⇔ A[i, j] = 1

v1

v2v3 v4 A =

0 1 1 0

1 0 1 1

1 1 0 0

0 1 0 0

A graph and its adjacency matrix.

• The degree degree, d(v)of a vertex v of G, denoted by d(v) or deg(v), is the number ofedges incident to v.

v1

v2v3 v4

deg(v1) = 2, deg(v2) = 3, deg(v3) = 2, deg(v4) = 1

• A vertex of degree 1 in G is called a leaf leaf, and a vertex of degree 0 in G iscalled an isolated vertex isolated vertex.

• The degree sequence degree sequenceof G is the multiset of degrees of vertices of G, e.g. inthe example above the degree sequence is {1, 2, 2, 3}.

• The minimum degree of G minimumdegree, δ(G)

, denoted by δ(G), is the smallest vertex degree inG (it is 1 in the example).

• The maximum degree of G maximumdegree, ∆(G)

, denoted by ∆(G), is the highest vertex degree inG (it is 3 in the example).

• G is called k-regular regularfor a natural number k if all vertices have degree k.Graphs that are 3-regular are also called cubic cubic.

7

1. preliminaries

• The average degree of G average degree,d(G)

is defined as d(G) =(∑v∈V deg(v)

)/|V|. Clearly,

we have δ(G) ≤ d(G) ≤ ∆(G) with equality if and only if G is k-regularfor some k.

Lemma 1 (Handshake lemma, 1.2.11.2.1). For every graph G = (V, E) we have

2|E| = ∑v∈V

d(v).

Corollary 2. In particular, the sum of all vertex degrees is even and therefore thenumber of vertices with odd degree is even.

Subgraphs

Definition.

• A graph H = (V′, E′) is a subgraph of G subgraph, ⊆, denoted by H ⊆ G, if V′ ⊆ V andE′ ⊆ E. In particular, G1 = G2 if and only if G1 ⊆ G2 and G2 ⊆ G1.

v1

v2v3 v4

v1

v2v3⊆

• A subgraph H of G is called an induced subgraph inducedsubgraph

of G if for every twovertices u, v ∈ V(H) we have uv ∈ E(H) ⇔ uv ∈ E(G). In the exampleabove H is not an induced subgraph of G. Every induced subgraph of Gcan be obtained by deleting vertices (and all incident edges) from G.

Examples:

v1

v2v3 v4

v1

v2v3

v1

v2 v4 v3 v4v2v3 v4

v1

• Every induced subgraph of G is uniquely defined by its vertex set. Wewrite G[X] G[X]for the induced subgraph of G on vertex set X, i.e. G[X] =(

X,{

xy : x, y ∈ X, xy ∈ E(G)})

. Then G[X] is called the subgraph of Ginduced by the vertex set X ⊆ V(G).

Example:

4 1

2

3 5 4 1

2

3

G G[{1, 2, 3, 4}

]

• A subgraph H = (V′, E′) of G = (V, E) is called a spanning subgraph spanningsubgraph

of Gif V′ = V.

• G is called bipartite bipartiteif there exists natural numbers m, n such that G ⊆ Km,n.

8

1. preliminaries

• A cycle (path, clique, independent set) in G is a subgraph H of G that isisomorphic to a cycle (path, clique, independent set).

• A walk walk(of length k) is a non-empty alternating sequence v0e0v1e1 · · · ek−1vkof vertices and edges in G such that ei = {vi, vi+1} for all i < k. If v0 = vk,the walk is closed closed walk.

• Let A, B ⊆ V, A ∩ B = ∅. A path P in G is called an A-B-path A-B-pathif P =v1 . . . vk, V(P) ∩ A = {v1} and V(P) ∩ B = {vk}. When A = {a} andB = {b}, we simply call P an a-b-path. If G contains an a-b-path we saythat the vertices a and b are linked by a path.

• Two paths P, P′ in G are called independent independentpaths

if every vertex in both P and P′

is an endpoint of P and P′.

• G is called connected connectedif any two vertices are linked by a path.

• A maximal connected subgraph of G is called a connected component componentof G.

• G is called acyclic acyclicif G does not have any cycle. Acyclic graphs are alsocalled forests forest.

• G is called a tree treeif G is connected and acyclic.

Proposition 3. If a graph G has minimum degree δ(G) ≥ 2, then G has a path oflength δ(G) and a cycle with at least δ(G) + 1 vertices.

Proposition 4. If a graph has an u-v-walk, then it has an u-v-path.

Proposition 5. If a graph has a closed walk of odd length, then it contains anodd cycle.

Proposition 6. If a graph has a closed walk with a non-repeated edge (at least oneedge appears in the walk with multiplicity one), then the graph contains a cycle.

Proposition 7. A graph is bipartite if and only if it has no cycles of odd length.

Definition. An Eulerian tour of G Eulerian touris a closed walk containing all edges of G, eachwith multiplicity one.

Theorem 8 (Eulerian tour condition, 1.8.11.8.1). A connected graph has an Euleriantour if and only if every vertex has even degree.

Lemma 9. Every tree on at least two vertices has a leaf.

Lemma 10. A tree of order n ≥ 1 has exactly n − 1 edges.

Lemma 11. Every connected graph contains a spanning tree.

Lemma 12. A connected graph on n ≥ 1 vertices and n − 1 edges is a tree.

Lemma 13. The vertices of every connected graph can be ordered (v1, . . . , vn) so thatfor every i ∈ {1, . . . , n} the graph G

[{v1, . . . , vi}

]is connected.

9

1. preliminaries

Operations on graphs

Definition. Let G = (V, E) and G′ = (V′, E′) be two graphs, U ⊆ V be a subsetof vertices of G and F ⊆ (V

2) be a subset of pairs of vertices of G. Then we define

• G ∪ G′ := (V ∪ V′, E ∪ E′) G ∪ G′, G ∩ G′and G ∩ G′ := (V ∩ V′, E ∩ E′). Note thatG, G′ ⊆ G ∪ G′ and G ∩ G′ ⊆ G, G′. Sometimes, we also write G + G′ forG ∪ G′.

• G − U := G[V \ U], G − F := (V, E \ F) G − U, G − F,G + F

and G + F := (V, E ∪ F). IfU = {u} or F = {e} then we simply write G − u, G − e and G + e forG − U, G − F and G + F, respectively.

• For an edge e = xy in G we define G ◦ e G ◦ eas the graph obtained from G byidentifying x and y and removing (if necessary) loops and multiple edges.

xy

v3

v2

v1

v4

v5

vxyv2

v1

v3

v4

v5

• The complement of G complement, G, denoted by G or GC, is defined as the graph (V, (V2) \

E). In particular, G + G is a complete graph, and G = (G + G)− E.

More graph parameters

Definition. Let G = (V, E) be any graph.

• The girth of G girth, g(G), denoted by g(G), is the length of a shortest cycle in G. If Gis acyclic, its girth is said to be ∞.

• The circumference circumferenceof G is the length of a longest cycle in G. If G is acyclic,its circumference is said to be 0.

• G is called Hamiltonian Hamiltonianif G has a spanning cycle, i.e. there is a cycle in Gthat contains every vertex of G. In other words, G is Hamiltonian if andonly if its circumference is |V|.

• G is called traceable traceableif G has a spanning path, i.e. there is a path in G thatcontains every vertex of G.

• For two vertices u and v in G, the distance between u and v distance, d(u, v), denoted byd(u, v), is the length of a shortest u-v-path in G. If no such path exists,d(u, v) is said to be ∞.

10

1. preliminaries

• The diameter of G diameter,diam(G)

of G, denoted by diam(G), is the maximum distanceamong all pairs of vertices in G, i.e.

diam(G) = maxu,v∈V

d(u, v).

• The radius of G radius, rad(G)of G, denoted by rad(G), is defined as

rad(G) = minu∈V

maxv∈V

d(u, v).

• If there is a vertex ordering v1, . . . , vn of G for a d ∈ N such that

|N(vi) ∩ {vi+1, . . . , vn}| ≤ d,

for all i ∈ [n] then G is called d-degenerate d-degenerate. The minimum d for which G isd-degenerate is called the degeneracy degeneracyof G.

v1 v2 vnv3

≤ d ≤ d ≤ d

We remark that the 1-degenerate graphs are precisely the forests.

Proposition. For any graph G = (V, E) each of the following is equivalent.

(i) G is a tree, that is, G is connected and acyclic.

(ii) G is minimally connected.

(iii) G is maximally acyclic.

(iv) G is connected and 1-degenerate.

(v) G is connected and |E| = |V| − 1.

(vi) G is acyclic and |E| = |V| − 1.

(vii) G is connected and every non-trivial subgraph of G has a vertex ofdegree at most 1.

(viii) Any two vertices are joined by a unique path in G.

11

2. matchings

2 Matchings

Definition.

• A matching matching(independent edge set) is a vertex-disjoint union of edges.

. . .

• A matching in G is a subgraph of G isomorphic to a matching. We denotethe size of the largest matching in G by ν(G) ν(G).

• A vertex cover in G vertex coveris a set of vertices U ⊆ V such that each edge in E isincident to at least one vertex in U. We denote the size of the smallestvertex cover in G by τ(G) τ(G).

U

• A k-factor of G k-factoris a k-regular spanning subgraph of G.

• A 1-factor of G is also called a perfect matching perfect matchingsince it is a matching oflargest possible size in a graph of order |V|. Clearly, G can only contain aperfect matching if |V| is even.

• A k-edge colouring edge colouringis an assignment c′ : E → [k] of edges to colours in [k]such that no two edges incident to the same vertex receive the same colour.The chromatic index of G chromatic index,

χ′(G)is the minimal k such that G has a k-edge colouring.

It is denoted by χ′(G).

• A k-vertex colouring vertex colouringis an assignment c : V → [k] of vertices to colours in [k]such that no two adjacent vertices receive the same colour. The chromaticnumber of G chromatic

number, χ(G)is the minimal k such that G has a k-vertex colouring. It is

denoted by χ(G).

12

2. matchings

Theorem 14 (Hall’s marriage theorem 1935, 2.1.22.1.2). Let G be bipartite with partitesets A and B. Then G has a matching containing all vertices of A if and only if|N(S)| ≥ |S| for all S ⊆ A.

S

N(S)

S

N(S)

A

B

bad

Theorem 15 (Tutte’s theorem 1947, 2.2.12.2.1). For S ⊆ V define q(S) to be thenumber of odd components of G − S. A graph G has a perfect matching if andonly if q(S) ≤ |S| for all S ⊆ V.

S

odd

odd

odd

|S| ≥ odd components of G− S

Corollary 16.

• Let G be bipartite with partite sets A and B such that |N(S)| ≥ |S| − d for allS ⊆ A, and a fixed positive integer d. Then G contains a matching of size atleast |A| − d.

• A k-regular bipartite graph has a perfect matching.

• A k-regular bipartite graph has a proper k-edge coloring.

13

2. matchings

Definition.

• For all functions f : V → N a f -factor of G f -factoris a spanning subgraph H of Gsuch that degH(v) = f (v) for all v ∈ V.

• Let f : V → N be a function with f (v) ≤ deg(v) for all v ∈ V. We canconstruct the auxiliary graph T(G, f ) T(G, f )by replacing each vertex v with vertexsets A(v) ∪ B(v) such that |A(v)| = deg(v) and |B(v)| = deg(v)− f (v).For adjacent vertices u and v we place an edge between A(u) and A(v)such that the edges between the A-sets are independent. We also insert acomplete bipartite graph between A(v) and B(v) for each vertex v.

1 v2

1

3 v1

2

3 2

∅

B(v1) A(v1) B(v2)

A(v2)

→

• Let H be a graph. A H-factor of G H-factoris a spanning subgraph of G that is avertex-disjoint union of copies of H.

H = G =

Lemma 17. Let f : V → N be a function with f (v) ≤ deg(v) for all v ∈ V. G has af -factor if and only if T(G, f ) has a 1-factor.

Theorem 18 (König’s theorem 1931, 2.1.12.1.1). Let G be bipartite. Then ν(G) = τ(G),i.e. the size of a largest matching is the same as the size of a smallest vertexcover.

Theorem (Hajnal and Szemerédi 1970). If G satisfies δ(G) ≥ (1 − 1/k)n, where k is adivisor of n, then G has a Kk-factor.

Theorem (Alon and Yuster 1995). Let H be a graph. If G satisfies

δ(G) ≥(

1 − 1χ(H)

)n,

then G contains at least(1 − o(1)

)· n/|V(H)| vertex-disjoint copies of H.

14

3. connectivity

3 Connectivity

Definition.

• For a natural number k ≥ 1, a graph G is called k-connected k-connectedif |V(G)| ≥ k+ 1and for any (k − 1)-set U of vertices in G the graph G − U is connected.

• The maximum k for which G is k-connected is called the connectivity of G connectivity,κ(G)

,denoted by κ(G).

κ(v1

v2v3 v4) = 1, κ(Cn) = 2, κ(Kn,m) = min{m, n}.

• For a natural number k ≥ 1, a graph G is called k-linked k-linkedif for any 2kdistinct vertices s1, s2, . . . , sk, t1, t2, . . . , tk there are vertex-disjoint si-ti-paths,i = 1, . . . , k.

s1

s2

s3

t1

t2

t3

s1

s2

s3

t′1 = t2

t′2 = t3

t′3 = t1

• If G is connected, but G − X is disconnected for a subset X of vertices of G,then X is called a cut set cut setof G. If a cut set consists of a single vertex v, thenv is called a cut vertex cut vertexof G.

• For a natural number ℓ ≥ 1, a graph G is called ℓ-edge-connected ℓ-edge-connected

if E(G) = ∅and for any (ℓ− 1)-set F of edges in G the graph G − F is connected.

• The edge-connectivity of G edge-connectivity,κ′(G)

is the maximum ℓ such that G is ℓ-edge-connected.It is denoted by κ′(G) or λ(G).

G non-trivial tree ⇒ λ(G) = 1, G cycle ⇒ λ(G) = 2.

• If G is connected, but G − e is disconnected for an edge e of G, then e iscalled a cut edge or bridge cut edge, bridgeof G.

Clearly, for every k, ℓ ≥ 2 if a graph is k-connected, k-linked or ℓ-edge-connected, thenit is also (k − 1)-connected, (k − 1)-linked or (ℓ− 1)-edge-connected, respectively.Moreover, for a non-trivial graph is it equivalent to be 1-connected, 1-linked, 1-edge-connected, or connected.

Lemma 19. For any connected, non-trivial graph G we have

κ(G) ≤ λ(G) ≤ δ(G).

15

3. connectivity

K100K100

A graph G with κ(G), λ(G) ≪ δ(G).

Definition. For a subset X of vertices and edges of G and two vertex sets A, B wesay that X separates A and B separateif each A-B-path contains an element of X.

A B

v3v2v1

u1 u2u3

e1 e2

e3 e4

e5

Some sets separating A and B: {e1, e4, e5}, {e1, u2}, {u1, u3, v3}

Note that a separating set of vertices must contain A ∩ B.

Theorem 20 (Menger’s theorem 1927, 3.3.13.3.1). For any graph G and any vertex setA, B ⊆ V(G) we have

min #vertices separating A and B = max #independent A-B-paths.

Corollary 21. If a, b are vertices of G, {a, b} /∈ E(G), then

min #vertices separating a and b = max #independent a-b-paths

a b

A B

Theorem 22 (Global version of Menger’s theorem, 3.3.63.3.6). A graph G is k-connected if and only if for any two vertices a, b in G there exist k independenta-b-paths.

Note that Menger’s Theorem implies that if G is k-linked, then G is k-connected.Moreover, Bollobás and Thomason proved in 1996 that if G is 22k-connected, then Gis k-linked.

Definition. The line graph L(G) line graph L(G)of G is the graph L(G) = (E, E′), where

E′ ={{e1, e2} ∈

(E2

): e1 incident to e2 in G

}.

16

3. connectivity

v2v3

v4

v1

e2e1

e3 e4 e5

e1

e2

e3

e5e4

A graph and its line graph.

Corollary. If a, b are vertices of G, then

min #edges separating a and b = max #edge-disjoint a-b-paths

a b

AB

Moreover, a graph is k-edge-connected if and only if there are k edge-disjoint pathsbetween any two vertices.

Definition. An ear-decomposition ear-decomposition

of a graph G is a sequence G1 ⊆ G2 ⊆ · · · ⊆ Gkof graphs, such that

• G1 is a cycle

• for each i = 2, . . . , k the graph Gi is given by Gi−1 + Pi, where Pi, called anear ear, is a Gi-path

• Gk = G

Gi

Theorem 23. A graph is 2-connected if and only if it has an ear-decomposition.

Lemma. If G is 3-connected, then there exists an edge e of G such that G ◦ e is also3-connected.

17

3. connectivity

Theorem 24. A graph G is 3-connected if and only if there exists a sequence ofgraphs G0, G1, . . . , Gn, such that

• G0 = K4

• for each i = 2, . . . , k the graph Gi has two adjacent vertices x′, x′′ of degreeat least 3, so that Gi = Gi+1 ◦ x′x′′

• Gn = G

xx′

x′′

y y′

y′′

Definition. The block-cut-vertex graph or block graph block-cut-vertexgraph

of G is a bipartite graph Hwhose partite sets are the blocks of the G, the bridges and maximal 2-connectedsubgraphs of G, and the cut vertices of G. There is an edge between a block anda cut vertex if and only if the block contains the cut vertex.

B1 B2 B3 B4

B5

B6

v1 v2 v3

v4

v5

B1 B2

v1 v2

...

The leaves of this graph are called block leaves block leaf.

Theorem 25. The block-cut-vertex graph of a connected graph is a tree.

18

4. planar graphs

4 Planar graphs

This section deals with graph drawings. We restrict ourselves to graph drawingsin the plane R2. It is also feasible to consider graph drawings in other topologicalspaces, such as the torus R2/Z2.

Definition.

• The line segment line segmentbetween p ∈ R2 and q ∈ R2 is the set {p + λ(q − p) : 0 ≤λ ≤ 1}.

• A homeomorphism homeomor-phism

is a continuous function that has a continuous inversefunction.

• Two sets A ∈ R2 and B ∈ R2 are said to be homeomorphic homeomorphicif there is ahomeomorphism f : A → B.

• An arc arcin R2 is a homeomorphic image of a line segment.

• A set A ⊆ R2 is path-connected path-connectedif there is a continuous path between anytwo points in A.

• Let A ⊆ R2 be a set of points. A region of A regionis a maximal path-connectedsubset R of A. Its boundary δR is also called its frontier frontier.

• A polygon polygonis a union of finitely many line segments that is homeomorphicto the circle S1 := {x ∈ R2 : ∥x∥ = 1}

• A plane graph plane graphG = V ∪ E is a subset of R2 consisting of a set of vertices Vand a set of edges E such that

1. V is a finite set of points in R2,

2. E is a finite set of arcs between vertices,

3. different edges have different endpoints,

4. the interior of an edge contains no vertex and no points of anotheredge.

The regions of R2 \ G, denoted by F(G), are called the faces of G faces, F(G). Faces withthree vertices are called triangles triangle. If all of the faces in F(G) are triangles,then G is called a plane triangulation triangulation.

• A plane graph is maximally plane maximally planeif one cannot add edges and still obtain aplane graph.

19

4. planar graphs

• G = (V, E) is planar planar graphif it has a plane embedding, i.e. if there is a planegraph G′ = V′ ∪ E′ and a bijection f : V → V′ such that uv ∈ E if and onlyif G′ has an edge between f (u) and f (v).

• G = (V, E) is outerplanar outerplanargraph

if it has a plane embedding such that the boundaryof the outer face contains all of the vertices V.

Theorem (Fáry’s theorem). Every planar graph has a plane embedding with straightline segments as edges.

Lemma (Jordan curve theorem). Let P ⊆ R2 be a polygon. Then R2 \ P has exactlytwo regions. One of the regions is unbounded, the other is bounded. Each of the tworegions has P as frontier.

Lemma. Let P1, P2 and P3 be internally disjoint arcs that have the same endpoints.Then

1. R2 \ (P1 ∪ P2 ∪ P3) has exactly three regions with boundaries P1 ∪ P2, P1 ∪ P3and P2 ∪ P3, respectively.

2. Let P be an arc from the interior of P1 to the interior of P3 whose interior liesin the region of R2 \ (P1 ∪ P3) containing the interior of P2. Then P contains apoints of P2.

P1 P2P3

P

Lemma. Let G be a plane graph and e an edge of G. Then

• A frontier X of a face of G either contains e or is disjoint to the interior of e.

• If e is on a cycle in G, then e is on the frontier of exactly two faces.

• If e is on no cycle in G, then e is on the frontier of exactly one face.

Lemma 26. A plane graph is maximally plane if and only if each of its faces is atriangle.

20

4. planar graphs

Theorem 27 (Euler’s formula, 4.2.94.2.9). Let G be a connected plane graph withv vertices, e edges and f faces. Then

v − e + f = 2.

Corollary. Let G = V ∪ E be a plane graph. Then

• |E| ≤ 3|V| − 6 with equality exactly if G is a plane triangulation.

• |E| ≤ 2|V| − 4 if no face in F(G) is a triangle.

Lemma (Pick’s formula). Let P be a polygon with corners on the grid Z2, A its area,I the number of grid points inside of P and B be the number of grid points on theboundary of P. Then A = I + B/2 − 1.

Definition. Let G and X be two graphs.

• We say that X is a minor of G minor, G = MX, denoted by G = MX, if X can be obtainedfrom G by successive vertex deletions, edge deletions and edge contrac-tions.

G

X

G = MX

G X

G = MX

An alternative characterisation is that the vertices V′ of a subgraph of G canbe partitioned into sets V′ = V1 + · · ·+V|V(X)| such that G[Vi] is connectedand Vi ∼ Vj if and only if the corresponding vertices vi, vj ∈ V(X) areadjacent for each i, j ∈ [|V(X)|].

X

G = MX

• X is a single-edge subdivision of G if V(X) = V(G) ∪ {v} and E(X) =E(G)− xy + xv + vy for xy ∈ E(G) and v ∈ V(G). X is a subdivision of G subdivisionif it can be obtained from G by a series of single-edge subdivisions.

21

4. planar graphs

• We say that X is a topological minor of G topologicalminor, G = TX

, denoted by G = TX, if a subgraphof G is a subdivision of X.

G XG = TX

G = TK4

Theorem 28 (Kuratowski’s theorem 1930, 4.4.64.4.6). A graph is planar if and only ifit does not have K5 or K3,3 as topological minors.

Definition.

• Let X be a set and ≤ ⊆ X2 a relation on X. Then ≤ is a partial order partial orderif itis reflexive, antisymmetric and transitive. A partial order is total total orderif x ≤ yor y ≤ x for every x, y ∈ X.

• Let ≤ be a partial order on a set X. The pair (X,≤) is called a poset poset(partially ordered set). If ≤ is clear from context, the set X itself is calleda poset. The poset dimension of (X,≤) poset dimension,

dim(X,≤)is the smallest number d such that

there are total orders R1, . . . , Rd on X with ≤ = R1 ∩ · · · ∩ Rd.

dim( ) = 1, dim(x y ) = 2 since x y = yx ∩ x

y

• The incidence poset (V ∪ E,≤) incidence poseton a graph G = (V, E) is given by v ≤ e ifand only if e is incident to v for all v ∈ V and e ∈ E.

v1

v2

v3

v4

e1e2

e3

e1 e2 e3

v1 v2 v3 v4

Theorem (Schnyder). Let G be a graph and P be its incidence poset. Then G is planarif and only if dim(P) ≤ 3.

Theorem 29 (5-coloring theorem, 5.1.25.1.2). Every planar graph is 5-colorable.

The more well-known 4-coloring theorem is much harder to prove. Interestingly, itis one of the first theorems that has been proved using computer assistance. Thecomputer-generated proof uses an enormous case distinction. Some mathematicianshave philosophical problems with this approach since the resulting proof cannot beeasily verified by humans. A shorter proof is still outstanding.

Theorem (4-coloring theorem). Every planar graph is 4-colorable.

22

4. planar graphs

Definition.

• Let L(v) ⊆ N be a list of colors for each vertex v ∈ V. We say that Gis L-list-colorable L-list-colorableif there is coloring c : V → N such that c(v) ∈ L(v) foreach v ∈ V and adjacent vertices receive different colors.

• Let k ∈ N. We say that G is k-list-colorable or k-choosable k-list-colorableif G is L-list-colorable for each list L with |L(v)| = k for all v ∈ V.

• The choosability choosability,ch(G)

, denoted by ch(G), is the smallest k such that G is k-choosable.

• The edge choosability edgechoosability,ch′(G)

, denoted by ch′(G), is defined analogously.

Theorem 30 (Thomassen’s theorem 1994, 5.4.25.4.2). Every planar graph is 5-choosable.

23

5. colorings

5 Colorings

Lemma (Greedy estimate for the chromatic number).Let G be graph. Then χ(G) ≤ ∆(G) + 1.

Theorem 31 (Brook’s theorem 1924, 5.2.45.2.4). Let G be a connected graph.Then χ(G) ≤ ∆(G) unless G is a complete graph or an odd cycle.

Definition.

• The clique number ω(G) of G clique number,ω(G)

is the largest order of an induced completesubgraph of G.

• The co-clique number α(G) of G co-cliquenumber, α(G)

is the largest order of an induced emptysubgraph of G.

• A graph G is called perfect perfect graphif χ(H) = ω(H) for each induced subgraph Hof G. For example, bipartite graphs are perfect with χ = ω = 2.

Lemma (Small coloring results).

• χ(G) ≥ max{ω(G), n/α(G)} since each color class is an empty induced sub-graph and χ(Kk) = k.

• ∥G∥ ≥ (χ(G)2 ) ⇔ χ(G) ≤ 1/2 +

√2∥G∥+ 1/4 since there must be at least one

edge between any two color classes.

• The chromatic number χ(G) of G is at most one more than the length of alongest directed path in any orientation of G.

Theorem (Perfect graph theorem 1972). A graph G is perfect if and only if itscomplement G is perfect.

Theorem (Strong perfect graph theorem 2002). A graph G is perfect if and only if itdoes not contain an odd cycle on at least 5 vertices (an odd hole) or the complement ofan odd hole as an induced subgraph.

Definition. Let A be the adjacency matrix of a graph G.

• By the spectral theorem the symmetric matrix A has an orthonormal basisof eigenvectors and all of its eigenvalues are real.

• The spectrum λ(G) of G spectrum, λ(G)is the multiset of eigenvalues of A.

• The spectral radius of G spectral radius,λmax(G)

is λmax(G) := max{

λ : λ ∈ λ(G)}

.Analogously, λmin(G) := min

{λ : λ ∈ λ(G)

}.

Lemma (Small results about the eigenvalues of G). Let A be the adjacency matrixof G and let H be an induced subgraph of G. Then

• λmin(G) ≤ λmin(H) ≤ λmax(H) ≤ λmax(G),

24

5. colorings

• δ(G) ≤ 2∥G∥/n ≤ λmax(G) ≤ ∆(G),

• trace(A) = 0, trace(A2) = 2∥G∥, trace(A3) = 6 · # triangles in G.

Theorem 32 (Spectral estimate for the chromatic number from folklore).Let G be a graph. Then χ(G) ≤ λmax(G) + 1.

Example (Mycielski’s construction).We can construct a family

(Gk = (Vk, Ek)

)k∈N

of triangle-free graphs with χ(Gk) = kas follows:

• G1 is the single-node graph, G2 is the single-edge graph.

• Vk+1 := Vk ∪ U ∪ {w} where Vk ∩(U ∪ {w}

)= ∅, V = {v1, . . . , vn} and

U = {u1, . . . , un}.

• Ek+1 := Ek ∪{

wui : i = 1, . . . , k}∪⋃n

i=1{

uiv : v ∈ NGk (vi)}

.

G1 G2 G3

Example (Tutte’s construction). We can construct a family (Gk)k∈N of triangle-freegraphs with χ(Gk) = k as follows: G1 is the single-node graph. To get from Gk toGk+1, take an independent set U of size k(|Gk| − 1) + 1 and ( |U|

|Gk |) copies of Gk. Foreach subset of size |Gk| in U then introduce a perfect matching to exactly one of thecopies of Gk.

G1

U

G2 G3

(32) · G2

Theorem 33 (König’s theorem 1916, 5.3.15.3.1).Let G be a bipartite graph. Then χ′(G) = ∆(G).

Theorem 34 (Vizing’s theorem 1964, 5.3.25.3.2).Let G be a graph. Then χ′(G) ∈ {∆(G), ∆(G) + 1}.

Thus, the chromatic index of a graph can only take one of two possible values.Determining which of the two values occurs is NP-complete.

Lemma. We have ch(Kn,n) ≥ c · log(n) for some constant c > 0. In particular,

ch(

K(2k−1

k ),(2k−1k )

)≥ c · k.

Theorem 35 (Galvin’s theorem 1995, 5.4.45.4.4).Let G be a bipartite graph. Then ch′(G) = χ′(G).

25

6. extremal graph theory

6 Extremal graph theory

In this section c, c1, c2, . . . always denote unspecified constants in R>0.

Definition.

• Let n be a positive integer and H a graph. By ex(n, H) ex(n, H)we denote themaximum size of a graph of order n that does not contain H as a subgraph.EX(n, H) EX(n, H)is the set of such graphs.

• Let n and r be integers with 1 ≤ r ≤ n. The Turàn graph T(n, r) Turàn graph,T(n, r)

is thecomplete r-partite graph of order n whose partite sets differ by at most 1in size. It does not contain Kr+1. We denote ∥T(n, r)∥ by t(n, r) t(n, r).

T(5, 2) T(6, 2) T(7, 3)

Example.

• ex(n, K2) = 0, EX(n, K2) = {En}• ex(n, P3) = ⌊n/2⌋, EX(n, P3) = {⌊n⌋ · K2 + (n mod 2) · E1}

. . .H = EX(n, H)

Lemma (On Turàn graphs).

• Among all r-partite graphs on n vertices the Turàn graph T(n, r) has the largestnumber of edges.

• We have the recursion

t(n, r) = t(n − r, r) + (n − r)(r − 1) +(

r2

).

• A Turàn graph lacks a ratio of 1/r of the edges:

limn→∞

t(n, r)(n

2)=

(1 − 1

r

).

Theorem 36 (Turàn’s theorem 1941, 7.1.17.1.1). For all integers r > 1 and n ≥ 1,every graph G with n vertices, ex(n, Kr) edges and Kr ⊆ G is a Tr−1(n).

26

6. extremal graph theory

Definition. Let X, Y ⊆ V(G) be vertex sets and ϵ > 0.

• We define ∥X, Y∥ to be the number of edges between X and Y and thedensity d(X, Y) of (X, Y) density, d(X, Y)to be

d(X, Y) :=∥X, Y∥|X||Y| .

• For ϵ > 0 the pair (X, Y) is an ϵ-regular pair ϵ-regular pairif we have |d(X, Y)− d(A, B)| ≤ϵ for all A ⊆ X, B ⊆ Y with |A| ≥ ϵ|X| and |B| ≥ ϵ|Y|.

X YA B

• An ϵ-regular partition ϵ-regularpartition

of the graph G = (V, E) is a partition of the vertexset V = V0 ⊔ V1 ⊔ · · · ⊔ Vk with the following properties:

1. |V0| ≤ ϵ|V|2. |V1| = |V2| = · · · = |Vk|3. All but at most ϵk2 of the pairs (Vi, Vj) for 1 ≤ i < j ≤ k are ϵ-regular.

V0Vk

Vi

Vj

Theorem 37 (Erdos-Stone theorem 1946, 7.1.27.1.2). For all integers r > s ≥ 1 andevery ϵ > 0 there exists an integer n0 such that every graph with n ≥ n0 verticesand at least

tr−1(n) + ϵn2

edges contains Krs as a subgraph.

27

6. extremal graph theory

Theorem (Chvátal-Szemerédi theorem 1981). Chvátal and Szemerédi proved a morequantitative version of the Erdos-Stone theorem: For every ϵ > 0 and every integerr ≥ 3, every graph on n vertices and at least

(1 − 1/(r − 1) + ϵ

)(n

2) edges contains Ktr

as a subgraph. Here t is given by

t =log n

500 · log(1/ϵ).

Furthermore, there is a graph G on n vertices and(1 − (1 + ϵ)/(r − 1)

)(n

2) edges thatdoes not contain Kt

r for

t =5 · log n

log(1/ϵ),

i.e. the choice of t is asymptotically tight.

Definition. The Zarankiewicz Zarankiewicz,z(m, n; s, t)

function z(m, n; s, t) denotes the maximum numberof edges that a bipartite graph with parts of size m and n can have withoutcontaining Ks,t.

X

Y

m

n

s

t

forbidden

Theorem 38 (Kovári-Sós-Turán theorem 1954).We have the upper bound

z(m, n; s, t) ≤ (s − 1)1/t(n − t + 1)m1−1/t + (t − 1)m

for the Zarankiewicz function. In particular,

z(n, n; t, t) ≤ c1 · n · n1−1/t + c2 · n = O(n2−1/t)

for m = n and t = s.

Corollary.For t ≥ s ≥ 1 we can bound the extremal number of Kt,s using the Kovári–Sós–Turántheorem

ex(n, Kt,s) ≤12· z(n, n; s, t) ≤ cn2−1/s.

For t = s = 2 this bound yields

ex(n, C4) ≤n4(1 +

√4n − 3).

This bound is actually tight, i.e. ex(n, C4) = 1/2 · n3/2 · (1 + o(1)).

28

6. extremal graph theory

Lemma. ex(n, Kr,r) ≥ cn2−2/(r+1) for all n, r ∈ N.

Theorem 39. For all n ∈ N we have ex(n, Pk+1) ≤(n · (k − 1)

)/2.

Theorem 40 (Szemerédi’s regularity lemma 1970, 7.4.17.4.1). For every ϵ > 0 andevery integer m ≥ 1 there is an M ∈ N such that every graph of order at least mhas an ϵ-regular partition V0 ⊔ · · · ⊔ Vk with m ≤ k ≤ M.

Corollary 41. Erdos-Stone together with limn→∞ t(n, r)/(n2) = 1 − 1/r yields an

asymptotic formula for the extremal number of every graph H on at least one edge:

limn→∞

ex(n, H)

(n2)

=χ(H)− 2χ(H)− 1

For example, ex(n, ) ≃ 2/3 · (n2) since χ( ) = 4.

Conjecture (Hadwiger conjecture). Let r be a natural number and G be a graph.Then χ(G) ≥ r implies MKr ⊆ G.

For r ∈ {1, 2, 3, 4} this is easy to see. For r ∈ {5, 6} the conjecture has been provenusing the 4-color-theorem. It is still open for r ≥ 7.

Theorem 42. Every graph G of average degree at least cr2 contains Kr as a topologicalminor.

Theorem. Let G be a graph of minimum degree δ(G) ≥ d and girth g(G) ≥ 8k + 3for d, k ∈ N and d ≤ 3. Then G has a minor H of minimum degree δ(H) ≥ d(d − 1)k.

Theorem 43 (Thomassen’s theorem 1983, 7.2.57.2.5). For all r ∈ N there exists a functionf : N → N such that every graph of minimum degree at least 3 and girth at least f (r)has a Kr minor.

Theorem (Kühn-Osthus 2002). Let r ∈ N. Then there is a constant g ∈ N such thatwe have TKr ⊆ G for every graph G with δ(G) ≥ r − 1 and g(G) ≥ g.

29

7. ramsey theory

7 Ramsey theory

In every 2-coloring in this section we use the colors red and blue.

Definition.

• In an edge-coloring of a graph, a set of edges is

– monochromatic monochromaticif all edges have the same color,

– rainbow rainbowif no two edges have the same color,

– lexical lexicalif two edges have the same color if and only if they have thesame lower endpoint in some ordering of the vertices.

• Let k be a natural number. Then the Ramsey number R(k) ∈ N ∪ {∞} Ramsey, R(k)isthe smallest n such that every 2-edge-coloring of Kn contains a monochro-matic Kk.

Color E(Kn) in 2 colors.

or

• Let k and l be natural numbers. Then the asymmetric Ramsey number R(k, l) asymmetricRamsey, R(k, l)is the smallest n ∈ N ∪ {∞} such that every 2-edge-coloring of a Kn

contains a red Kk or a blue Kl .

• Let G and H be graphs. Then the graph Ramsey number R(G, H) graph Ramsey,R(G, H)

is thesmallest n ∈ N ∪ {∞} such that every 2-edge-coloring of Kn contains ared G or a blue H.

• Let r, l1, . . . , lk be natural numbers. Then the hypergraph Ramsey numberRr(l1, . . . , lk) hypergraph

Ramsey,Rr(l1, . . . , lk)

is the smallest n ∈ N ∪ {∞} such that for every k-coloringof ([n]r ) there is an i ∈ {1, . . . , k} and a V ⊆ [n] with |V| = li such that (V

r )has color i.

• Let G and H be graphs. Then the induced Ramsey number Rind(G, H) induced Ramsey,Rind(G, H)

is thesmallest n ∈ N ∪ {∞} such that there is a graph F on n vertices every2-coloring of which contains a red G or a blue H.

• For n ∈ N and a graph H, the anti-Ramsey number AR(n, H) anti-Ramsey,AR(n, H)

is the max-imum number of colors that an edge-coloring of Kn can have withoutcontaining a rainbow copy of H.

30

7. ramsey theory

Lemma.

• R(3) = 6, i.e. every 2-edge-colored K6 contains a monochromatic triangle andthere is a 2-coloring of a K5 without monochromatic triangles.

• Clearly, R(2, k) = R(k, 2) = k.

Theorem 44 (Ramsey theorem 1930, 9.1.19.1.1). For every k ∈ N we have√

2k ≤

R(k) ≤ 4k. In particular, the Ramsey numbers, the asymmetric Ramsey numbersand the graph Ramsey numbers are finite.

Theorem 45. For every k, l ∈ N we have R(k, l) ≤ R(k − 1, l) + R(k, l − 1). Thisimplies R(k, l) ≤ (k+l−2

k−1 ) by induction.

Lemma 46. For every r, p, q ∈ N we have Rr(p, q) ≤ Rr−1(

Rr(p − 1, q), Rr(p, q −1))+ 1.

Lemma. We have c1 · 2k ≤ R2(3, . . . , 3︸ ︷︷ ︸k

) ≤ c2 · k! for some constants c1, c2 > 0.

Applications of Ramsey theory

Theorem (Erdos-Szekeres 1935). Any sequence of (r − 1)(s − 1) + 1 distinct realnumbers contains an increasing subsequence of length r or a decreasing subsequenceof length s.

Theorem 47 (Erdos-Szekeres 1935). For every m ∈ N there is an N ∈ N such thatevery set of at least N points in R2 contains a convex m-gon.

Theorem (Schur 1916). Let c : N → [r] be a coloring of the natural numbers withr ∈ N colors. Then there are monochromatic x, y, z ∈ N with x + y = z.

Definition. Let r ∈ N and A ∈ Zn×k.

• A is said to be r-regular r-regular matrixif there is a monochromatic solution of Ax = 0 forevery r-coloring c : N → [r] of N.

31

7. ramsey theory

• A fulfils the column condition columncondition

if there is a partition C1 ⊔ · · · ⊔ Cl of thecolumns of A such that the following holds: Let si := ∑c∈Ci

c for i ∈ [l] bethe sum of columns in Ci. Then we must have s1 = 0 and every si for i ∈{2, . . . , l} is a rational linear combination of the columns in C1 ⊔ . . . ⊔ Ci−1.

For example, 2x1 + x2 + x3 − 4x4 fulfils the column condition since 2 + 1 +1 − 4 = 0.

Theorem 48 (Rado 1933). Let A ∈ Zn×k. If A fulfils the column condition, then A isr-regular for every r ∈ N.

Lemma 49. For every s, t ∈ N with s ≥ t ≥ 1 we have R(sK2, tK2) = 2s + t − 1.

t − 12s − 1

Lemma 50. For every s, t ∈ N with s ≥ t ≥ 1 and s ≥ 2 we have R(sK3, tK3) = 3s+ 2t.

Theorem 51 (Chvátal, Harary 1972). Let G and H be graphs. Then R(G, H) ≥(χ(G)− 1

)(c(H)− 1

)+ 1 where c(H) is the cardinality of the largest component

of H.

Kc(h)−1 χ(G)− 1

Theorem 52 (Induced Ramsey theorem).Rind(G, H) is finite for all graphs G and H.

Theorem 53 (Canonical Ramsey theorem, Erdos-Rado 1950). For all k ∈ N there isan n ∈ N such that any edge coloring of Kn with arbitrarily many colors contains aKk that is monochromatic, rainbow or lexical.

Theorem 54 (Chvátal-Rödl-Szemerédi-Trotter 1983). For every positive inte-ger ∆ there exists a c ∈ N such that R(H, H) ≤ c|V(H)| for every graph Hwith ∆(H) = ∆.

Corollary. For every n-vertex graph H with maximum degree 3 we have R(H, H) ≤cn for some constant c > 0. This number grows much slower than R(Kn, Kn) ≥

√2

n.

Theorem 55 (Anti-Ramsey theorem, Erdos-Simonvits-Sós 1973). For all n, r ∈ N wehave AR(n, Kr) = (n

2)(1 − 1/(r − 2)

)(1 − o(1)

).

32

8. flows

8 Flows

Definition. Let H be an Abelian semigroup and E := {(x, y) : xy ∈ E}.

• For f : E → H and X, Y ⊆ V we define f (X, Y) := ∑(x,y)∈(X×Y)∩E f (x, y).

• A function f : E → H is a circulation on G circulationif

(C1) f (x, y) = − f (y, x) for all xy ∈ E and

(C2) f (v, V) = 0 for all v ∈ V.x1

x2

x3

x4

−11

1 1

x1

x2

x3

x4

11

1 1

• If H is an Abelian group, then a circulation f is also called a H-flow on G H-flow.If f (x, y) = 0 for all xy ∈ E, then f is a nowhere-zero flow nowhere-zero.

1 −7

13

A nowhere-zero Z2-flow.

• For k ∈ N a k-flow k-flowis a Z-flow f such that 0 < | f (x, y)| < k for all xy ∈ E.The flow number φ(G) of G flow number,

φ(G)is the smallest k such that G has a k-flow.

• Let s ∈ V be a source, v ∈ V be a sink and c : E → Z≥0 be a capacity function.Then a network flow on the network (G, s, t, c) source, sink,

capacity,network flow

is f : E → R with the followingproperties for all x, y ∈ V:

(F1) f (x, y) = − f (y, x)

(F2) f (x, V) = 0 if x ∈ {s, t}(F3) f (x, y) ≤ c(x, y)

2

7

1

3

5

x1

x2

s t

f = 2f = 2

f = 3f = 3

f = 0

33

8. flows

For any S ⊆ V with s ∈ S and t ∈ S the pair (S, V \ S) is called a cut cut. Itscapacity is c(S, V \ S).

The value f (s, V) is also called the value of f value, | f |and is denoted by | f |.

Lemma.

• For any circulation f and X ⊆ V we have f (X, X) = 0, f (X, V) = 0 andf (X, V \ X) = 0

• For any network flow f and cut (S, S) we have f (S, S) = f (s, V).

Theorem 56 (Ford-Fulkerson theorem 1965, 6.2.26.2.2). In any network the maximumvalue of a flow is the same as the minimum capacity of a cut and there is anintegral flow f : E → Z≥0 with this maximum flow value.

Theorem 57 (Tutte 1954, 6.3.16.3.1). For every multigraph G there is a polynomial P ∈Z[X] such that for any finite Abelian group H the number of nowhere-zero H-flows on G is P

(|H| − 1

).

Corollary. If a H-flow on G exists for some finite Abelian group H, then there isalso a H-flow on G for all finite Abelian groups H with |H| = |H|. For example, if aZ4-flow exists, then a Z2 × Z2-flow also exists.

Theorem 58 (Tutte 1950, 6.3.36.3.3). A multigraph admits a k-flow if and only if it admitsa Zk-flow.

Theorem 59 (Tutte 1954, 6.5.36.5.3). For a planar graph G and its dual G∗ we haveχ(G) = φ(G∗).

Lemma. A graph has a 2-flow if and only if all of its degrees are even.

Lemma. A cubic (3-regular) graph has a 3-flow if and only if it is bipartite.

Conjecture (Tutte’s flow conjecture). Every bridgeless multigraph has flow numberat most 5. Seymore proved φ(G) ≤ 6 for bridgeless graphs in 1981.

34

9. random graphs

9 Random graphs

In this section we deal with randomly chosen graphs. We will often use the “proba-bilistic method”, a proof method for showing existence: By proving that an objectwith some desired properties can be chosen randomly (in some probability space)with non-zero probability, we also show that such an object exists.

Definition.

• G(n, p) is the probability space on all n-vertex graphs that results fromindependently deciding whether to include each of the (n

2) possible edgeswith fixed probability p ∈ [0, 1]. This model is called the Erdos–Rényi model Erdos–Rényiof random graphs.

• A property propertyP is a set of graphs, e.g. P = {G : G is k-connected}.

Let (pn) ∈ [0, 1]N be a sequence. We say that G ∈ G(n, pn) almost always almost alwayshas property P if Prob

(G ∈ G(n, pn

)∩ P) → 1 for n → ∞.

A function f (n) : N → [0, 1] is a threshold function thresholdfunction

for property P if:

– For all (pn) ∈ [0, 1]N with pn/ f (n) n→∞−→ 0 the graph G ∈ G(n, pn)almost always does not have property P .

– For all (pn) ∈ [0, 1]N with pn/ f (n) n→∞−→ ∞ the graph G ∈ G(n, pn)almost always has property P .

Lemma.

• For a given graph G on n vertices and m edges we have

Prob(G = G(n, p)

)= pm(1 − p)(

n2)−m

• For all integers n ≥ k ≥ 2 we have

Prob(G ∈ G(n, p), α(G) ≥ k

)≤

(nk

)(1 − p)(

k2)

and

Prob(G ∈ G(n, p), ω(G) ≥ k

)≤

(nk

)p(

k2)

Theorem 60 (Erdos 1947). Erdos proved the lower bound R(k, k) ≥ 2k/2 onRamsey numbers by applying the probabilistic method to the Erdos–Rényimodel.

35

9. random graphs

Lemma. We have

E(#k-cycles in G ∈ G(n, p)

)=

nk

2k· pk

where nk = n · (n − 1) · · · (n − k + 1).

Theorem 61 (Erdos 1959, 11.2.211.2.2). For every k ∈ N there is a graph H with g(H) ≥ kand χ(H) ≥ k.

Lemma. For all p ∈ (0, 1) and graphs H almost all graphs in G(n, p) contain H as aninduced subgraph.

Lemma. For all p ∈ (0, 1) and ϵ > 0 almost all graphs in G(n, p) fulfil

χ(G) >log

(1/(1 − p)

)

2 + ϵ· n

log n

Remark. Asymptotic behaviour of G(n, p) for some properties:

• pn =√

2/n2 ⇒ G almost always has a component with > 2 vertices

• pn = 1/n ⇒ G almost always has a cycle

• pn = log n/n ⇒ G is almost always connected

• pn = (1 + ϵ) log n/n ⇒ G almost always has a Hamiltonian cycle

• pn = n−2/(k−1) is the threshold function for containing Kk

Lemma 62 (Lovász local lemma). Let A1, . . . , An be events in some probabilisticspace. If Prob(Ai) ≤ p ∈ (0, 1), each Ai is independent from all but at most d ∈N of the other Ai and np(d + 1) ≤ 1, then

n⋂

i=1

Ai > 0.

Lemma. Van-der-Waerden’s number W(k) is the smallest n such that any 2-coloringof [n] contains a monochromatic arithmetic progression of length k. We can proveW(k) ≥ 2k−1/(ek2) with the Lovász local lemma.

1 2 3 4 5 6 7 8

36

10. hamiltonian cycles

10 Hamiltonian cycles

Lemma 63 (Necessary condition for Hamiltonian cycle). If G has a Hamiltoniancycle, then for every non-empty S ⊆ V the graph G − S cannot have morethan |S| components.

Non-hamiltonian graph.

Theorem 64 (Dirac 1952, 10.1.110.1.1). Every graph with n ≥ 3 vertices and minimumdegree at least n/2 has a Hamiltonian cycle.

K n2

K n2

δ = n/2 − 1

Theorem. Every graph on n ≥ 3 vertices with α(G) ≤ κ(G) is Hamiltonian.

Theorem 65 (Tutte 1956, 10.1.410.1.4). Every 4-connected planar graph is Hamiltonian.

Theorem 66 (Fleischner’s theorem 1974, 10.3.110.3.1). If G is 2-connected, then G2 :=(V, E′) with E′ := {uv : u, v ∈ V, dG(u, v) ≤ 2} is Hamiltonian.

Theorem 67 (Chvátal 1972, 10.2.110.2.1). Let 0 ≤ a1 ≤ · · · ≤ an < n be an integersequence with n ≥ 3. All graphs with the degree sequence a1, . . . , an areHamiltonian if and only if ai ≤ i implies an−i ≥ n − i for all i < n/2.

37

Literature

[1] Béla Bollobás. Modern Graph Theory. Graduate texts in mathematics. Heidelberg:Springer, 1998. isbn: 038-7-98488-7.

[2] J. A. Bondy and U. S. R. Murty. Graph Theory with Applications. New York:Elsevier, 1976. isbn: 978-0-444-19451-0.

[3] Gary Chartrand and Linda Lesniak. Graphs & Digraphs. Belmont, CA, USA:Wadsworth Publ. Co., 1986. isbn: 0-534-06324-1.

[4] Reinhard Diestel. Graph Theory, 4th Edition. Graduate texts in mathematics.Springer, 2012. isbn: 978-3-642-14278-9. url: http://diestel-graph-theory.comhttp://diestel-graph-theory.com

(cit. on p. 33).

[5] L. Lovasz. Combinatorial Problems and Exercises. Akadémiai Kiadó, 1979. isbn:978-0-821-84262-1.

[6] Douglas B. West. Introduction to Graph Theory. 2nd ed. Prentice Hall, Sept. 2000.isbn: 013-0-14400-2.

38

Named theorems1 Lemma (Handshake lemma, 1.2.11.2.1)1 Lemma (Handshake lemma, 1.2.11.2.1) . . . . . . . . . . . . . . . . . . . . . 88 Theorem (Eulerian tour condition, 1.8.11.8.1)8 Theorem (Eulerian tour condition, 1.8.11.8.1) . . . . . . . . . . . . . . . . . . 914 Theorem (Hall’s marriage theorem 1935, 2.1.22.1.2)14 Theorem (Hall’s marriage theorem 1935, 2.1.22.1.2) . . . . . . . . . . . . . . 1315 Theorem (Tutte’s theorem 1947, 2.2.12.2.1)15 Theorem (Tutte’s theorem 1947, 2.2.12.2.1) . . . . . . . . . . . . . . . . . . . 1318 Theorem (König’s theorem 1931, 2.1.12.1.1)18 Theorem (König’s theorem 1931, 2.1.12.1.1) . . . . . . . . . . . . . . . . . . . 1420 Theorem (Menger’s theorem 1927, 3.3.13.3.1)20 Theorem (Menger’s theorem 1927, 3.3.13.3.1) . . . . . . . . . . . . . . . . . 1622 Theorem (Global version of Menger’s theorem, 3.3.63.3.6)22 Theorem (Global version of Menger’s theorem, 3.3.63.3.6) . . . . . . . . . . 1627 Theorem (Euler’s formula, 4.2.94.2.9)27 Theorem (Euler’s formula, 4.2.94.2.9) . . . . . . . . . . . . . . . . . . . . . . 2128 Theorem (Kuratowski’s theorem 1930, 4.4.64.4.6)28 Theorem (Kuratowski’s theorem 1930, 4.4.64.4.6) . . . . . . . . . . . . . . . 2229 Theorem (5-coloring theorem, 5.1.25.1.2)29 Theorem (5-coloring theorem, 5.1.25.1.2) . . . . . . . . . . . . . . . . . . . . 2230 Theorem (Thomassen’s theorem 1994, 5.4.25.4.2)30 Theorem (Thomassen’s theorem 1994, 5.4.25.4.2) . . . . . . . . . . . . . . . 2331 Theorem (Brook’s theorem 1924, 5.2.45.2.4)31 Theorem (Brook’s theorem 1924, 5.2.45.2.4) . . . . . . . . . . . . . . . . . . . 2432 Theorem (Spectral estimate for the chromatic number from folklore)32 Theorem (Spectral estimate for the chromatic number from folklore) . 2533 Theorem (König’s theorem 1916, 5.3.15.3.1)33 Theorem (König’s theorem 1916, 5.3.15.3.1) . . . . . . . . . . . . . . . . . . . 2534 Theorem (Vizing’s theorem 1964, 5.3.25.3.2)34 Theorem (Vizing’s theorem 1964, 5.3.25.3.2) . . . . . . . . . . . . . . . . . . 2535 Theorem (Galvin’s theorem 1995, 5.4.45.4.4)35 Theorem (Galvin’s theorem 1995, 5.4.45.4.4) . . . . . . . . . . . . . . . . . . 2536 Theorem (Turàn’s theorem 1941, 7.1.17.1.1)36 Theorem (Turàn’s theorem 1941, 7.1.17.1.1) . . . . . . . . . . . . . . . . . . . 2637 Theorem (Erdos-Stone theorem 1946, 7.1.27.1.2)37 Theorem (Erdos-Stone theorem 1946, 7.1.27.1.2) . . . . . . . . . . . . . . . . 2738 Theorem (Kovári-Sós-Turán theorem 1954)38 Theorem (Kovári-Sós-Turán theorem 1954) . . . . . . . . . . . . . . . . 2840 Theorem (Szemerédi’s regularity lemma 1970, 7.4.17.4.1)40 Theorem (Szemerédi’s regularity lemma 1970, 7.4.17.4.1) . . . . . . . . . . . 2943 Theorem (Thomassen’s theorem 1983, 7.2.57.2.5)43 Theorem (Thomassen’s theorem 1983, 7.2.57.2.5) . . . . . . . . . . . . . . . 2944 Theorem (Ramsey theorem 1930, 9.1.19.1.1)44 Theorem (Ramsey theorem 1930, 9.1.19.1.1) . . . . . . . . . . . . . . . . . . 3147 Theorem (Erdos-Szekeres 1935)47 Theorem (Erdos-Szekeres 1935) . . . . . . . . . . . . . . . . . . . . . . . 3148 Theorem (Rado 1933)48 Theorem (Rado 1933) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3251 Theorem (Chvátal, Harary 1972)51 Theorem (Chvátal, Harary 1972) . . . . . . . . . . . . . . . . . . . . . . 3252 Theorem (Induced Ramsey theorem)52 Theorem (Induced Ramsey theorem) . . . . . . . . . . . . . . . . . . . . 3253 Theorem (Canonical Ramsey theorem, Erdos-Rado 1950)53 Theorem (Canonical Ramsey theorem, Erdos-Rado 1950) . . . . . . . . 3254 Theorem (Chvátal-Rödl-Szemerédi-Trotter 1983)54 Theorem (Chvátal-Rödl-Szemerédi-Trotter 1983) . . . . . . . . . . . . . 3255 Theorem (Anti-Ramsey theorem, Erdos-Simonvits-Sós 1973)55 Theorem (Anti-Ramsey theorem, Erdos-Simonvits-Sós 1973) . . . . . . 3256 Theorem (Ford-Fulkerson theorem 1965, 6.2.26.2.2)56 Theorem (Ford-Fulkerson theorem 1965, 6.2.26.2.2) . . . . . . . . . . . . . . 3457 Theorem (Tutte 1954, 6.3.16.3.1)57 Theorem (Tutte 1954, 6.3.16.3.1) . . . . . . . . . . . . . . . . . . . . . . . . . 3458 Theorem (Tutte 1950, 6.3.36.3.3)58 Theorem (Tutte 1950, 6.3.36.3.3) . . . . . . . . . . . . . . . . . . . . . . . . . 3459 Theorem (Tutte 1954, 6.5.36.5.3)59 Theorem (Tutte 1954, 6.5.36.5.3) . . . . . . . . . . . . . . . . . . . . . . . . . 3460 Theorem (Erdos 1947)60 Theorem (Erdos 1947) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3561 Theorem (Erdos 1959, 11.2.211.2.2)61 Theorem (Erdos 1959, 11.2.211.2.2) . . . . . . . . . . . . . . . . . . . . . . . . 3662 Lemma (Lovász local lemma)62 Lemma (Lovász local lemma) . . . . . . . . . . . . . . . . . . . . . . . . 3663 Lemma (Necessary condition for Hamiltonian cycle)63 Lemma (Necessary condition for Hamiltonian cycle) . . . . . . . . . . 3764 Theorem (Dirac 1952, 10.1.110.1.1)64 Theorem (Dirac 1952, 10.1.110.1.1) . . . . . . . . . . . . . . . . . . . . . . . . . 3765 Theorem (Tutte 1956, 10.1.410.1.4)65 Theorem (Tutte 1956, 10.1.410.1.4) . . . . . . . . . . . . . . . . . . . . . . . . . 3766 Theorem (Fleischner’s theorem 1974, 10.3.110.3.1)66 Theorem (Fleischner’s theorem 1974, 10.3.110.3.1) . . . . . . . . . . . . . . . 3767 Theorem (Chvátal 1972, 10.2.110.2.1)67 Theorem (Chvátal 1972, 10.2.110.2.1) . . . . . . . . . . . . . . . . . . . . . . . 37

39

Index=, see isomorphicA-B-path, 99AR(n, H), see anti-RamseyF(G), see facesG = MX, see minorG = TX, see topological minorG ∩ G′, 1010G ∪ G′, 1010G, see graphG + F, 1010G − F, 1010G − U, 1010G[X], see induced subgraphG ◦ e, 1010H-factor, 1414H-flow, 3333L-list-colorable, 2323N(v), see neighboursR(G, H), see graph RamseyR(k), see RamseyR(k, l), see asymmetric RamseyRr(l1, . . . , lk), see hypergraph RamseyRind(G, H), see induced RamseyT(G, f ), 1414T(n, r), see Turàn graph∆(G), see maximum degreeEX(n, H), 2626∥G∥, see size|G|, see order| f |, see valueα(G), see co-clique numberch′(G), see edge choosabilitych(G), see choosabilityχ′(G), see chromatic indexχ(G), see chromatic numberdeg(v), see degreeδ(G), see minimum degreediam(G), see diameterℓ-edge-connected, 1515ϵ-regular pair, 2727ϵ-regular partition, 2727ex(n, H), 2626κ′(G), see edge-connectivityκ(G), see connectivityλ(G), see spectrumλmax(G), see spectral radiusλ(G), see edge-connectivity

ν(G), see matchingω(G), see clique numberG, see complementdim(X,≤), see poset dimensionrad(G), see radius≃, see isomorphic⊆, see subgraphτ(G), see vertex coverφ(G), see flow numberd-degenerate, 1111d(G), see average degreed(X, Y), see densityd(u, v), see distanced(v), see degreef -factor, 1414g(G), see girthk-connected, 1515k-factor, 1212k-flow, 3333k-linked, 1515k-list-colorable, see k-list-colorabler-regular matrix, 3131t(n, r), 2626z(m, n; s, t), see Zarankiewicz

acyclic, 99adjacency matrix, 77adjacent, 44almost always, 3535anti-Ramsey, 3030arc, 44, 1919asymmetric Ramsey, 3030average degree, 88

bipartite, 88block leaf, 1818block-cut-vertex graph, 1818bridge, see cut edge

capacity, 3333choosability, 2323chromatic index, 1212chromatic number, 1212circulation, 3333circumference, 1010clique, see complete graphclique number, 2424

40

closed walk, 99co-clique number, 2424column condition, 3232complement, 1010complete bipartite graph, 55complete graph, 55component, 99connected, 99connected component, see componentconnectivity, 1515cubic, 77cut, 3434cut edge, 1515cut set, 1515cut vertex, 1515cycle, 55

degeneracy, 1111degree, 77degree sequence, 77density, 2727diameter, 1111directed graph, 44distance, 1010

ear, 1717ear-decomposition, 1717edge, 44edge choosability, 2323edge colouring, 1212edge-connectivity, 1515empty graph, 55Erdos–Rényi, 3535Eulerian tour, 99

faces, 1919factor, see k-factorflow number, 3333forest, 99frontier, 1919

girth, 1010graph, 44graph Ramsey, 3030

Hamiltonian, 1010homeomorphic, 1919homeomorphism, 1919hypercube, 66

hypergraph, 44hypergraph Ramsey, 3030

incidence poset, 2222incident, 44independent paths, 99independent set, see empty graphinduced Ramsey, 3030induced subgraph, 88isolated vertex, 77isomorphic, 44

Kneser graph, 66

leaf, 77lexical, 3030line graph L(G), 1616line segment, 1919

matching, 1212maximally plane, 1919maximum degree, 77minimum degree, 77minor, 2121monochromatic, 3030multigraph, 44

neighbours, 77network flow, 3333nowhere-zero, 3333

order, 77outerplanar graph, 2020

partial order, 2222path, 55path-connected, 1919perfect graph, 2424perfect matching, 1212Petersen graph, 66planar graph, 2020plane graph, 1919polygon, 1919poset, 2222poset dimension, 2222property, 3535

radius, 1111rainbow, 3030Ramsey, 3030

41

region, 1919regular, 77

separate, 1616sink, 3333size, 77source, 3333spanning subgraph, 88spectral radius, 2424spectrum, 2424subdivision, 2121subgraph, 88

threshold function, 3535topological minor, 2222

total order, 2222traceable, 1010tree, 99triangle, 1919triangulation, 1919Turàn graph, 2626

value, 3434vertex, 44vertex colouring, 1212vertex cover, 1212

walk, 99

Zarankiewicz, 2828

42