Lecture Notes Graph Theory by Prof. Dr. Maria Axenovich, Lecture notes of Mathematics

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Lecture Notes

Graph Theory

Prof. Dr. Maria Axenovich

December 6, 2016

Contents

• 1 Introduction
• 2 Notations
• 3 Preliminaries
• 4 Matchings
• 5 Connectivity
• 6 Planar graphs
• 7 Colorings
• 8 Extremal graph theory
• 9 Ramsey theory
• 10 Flows
• 11 Random graphs
• 12 Hamiltonian cycles
• References
• Index

3 Preliminaries

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

(V

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
• The edge e = {u, v} ∈

(V

2

is also denoted by e = uv.

• If e = uv ∈ E is an edge of G, then u is called adjacent to v and u is called adjacent, incident incident to e.
• If e 1 and e 2 are two edges of G, then e 1 and e 2 are called adjacent if e 1 ∩e 2 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 ∈ V we draw a point (or small disc) in the plane. And for each edge uv ∈ 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 = { 12 , 13 , 24 }

V = {A, B, C, D, E},

E = {AB, AC, AD, AE, CE}

A

B

C D

E

Definition (Graph variants).

• A directed graph is a pair G = (V, A) where V is a finite set and E ⊆ V 2. directed graph The 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 multigraph of elements from

(V

1

(V

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 ⊆ 2 X^ \ {∅}. hypergraph

Definition. For two graphs G 1 = (V 1 , E 1 ) and G 2 = (V 2 , E 2 ) we say that G 1 and G 2 are isomorphic , denoted by G 1 ' G 2 , if there exists a bijection φ : V 1 → V 2 with isomorphic, ' xy ∈ E 1 if and only if φ(x)φ(y) ∈ E 2. Loosely speaking, G 1 and G 2 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 G 1 = G 2 instead of G 1 ' G 2 whenever vertices =

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 complete graph, Kn

Kn on n vertices as the (unlabeled) graph isomorphic to ( [n],

([n] 2

. Complete graphs correspond to cliques.

K 5 K 3

• for( n ≥ 3, the cycle Cn on n vertices as the (unlabeled) graph isomorphic to cycle,^ Cn [n],

{i, i + 1} : 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 v 1 v 2 v 3 v (^5) v 4

v 6

C 6 = v 1 v 2 v 3 v 4 v 5 v 6 v 1

• the path Pn on n vertices as the (unlabeled) graph isomorphic to path,^ 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 its number of edges. We write Pn = 12... n.

1 2 3 4 5 6

• the empty graph En on n vertices as the (unlabeled) graph isomorphic to empty graph, En

[n], ∅

Empty graphs correspond to independent sets.

E 10

• for m ≥ 1, the complete bipartite graph complete bipartite graph, Km,n

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

n Km,n

(1,... , 1)

(0,... , 0)

weight 1 (# 1’s in a binary tuple)

weight 2

weight n-

n

(n 2 )

(n 3 )

(n n 2 )

( (^) n−n 1 )

Qn− 1 Qn−^1

0001

1001

Qn:

Basic graph parameters and degrees

Definition. 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 = ∅. non-trivial Equivalently, G is non-trivial if G is not an empty graph.
• The order of G , denoted by |G|, is the number of vertices of G, i.e., |G| = |V |. order, |G|
• The size of G , denoted by ‖G‖, is the number of edges of G, i.e., ‖G‖ = |E|. size,^ ‖G‖ Note that if the order of G is n, then the size of G is between 0 and

(n 2

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

, denoted by N (S), is the set of vertices in V that have an adjacent vertex in S. The elements of N (S) are called neighbours of S. Instead of N ({v}) for v ∈ V we usually write N (v). neighbour

• If the vertices of G are labeled v 1 ,... , vn, then there is an n × n matrix A with entries in { 0 , 1 }, which is called the adjacency matrix and is defined adjacency matrix as follows: vivj ∈ E ⇔ A[i, j] = 1

v 1 v 3 v 2 v (^4) A =

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. v 1 v 3 v 2 v 4 deg(v 1 ) = 2, deg(v 2 ) = 3, deg(v 3 ) = 2, deg(v 4 ) = 1

• A vertex of degree 1 in G is called a leaf , and a vertex of degree 0 in G is leaf called an isolated vertex. isolated vertex
• The degree sequence of G is the multiset of degrees of vertices of G, e.g. in degree sequence the example above the degree sequence is { 1 , 2 , 2 , 3 }.
• The minimum degree of G minimum degree, δ(G)

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

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

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

• 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
• 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-regular for some k.

Lemma 1 (Handshake Lemma, 1.2.1). For every graph G = (V, E) we have

2 |E| =

v∈V

d(v).

Corollary 2. The sum of all vertex degrees is even and therefore the number of vertices with odd degree is even.

Subgraphs

Definition.

• A graph H = (V ′, E′) is a subgraph of G , denoted by H ⊆ G, if V ′^ ⊆ V subgraph,^ ⊆ and E′^ ⊆ E. If H is a subgraph of G, then G is called a supergraph of H , supergraph,^ ⊇ denoted by G ⊇ H. In particular, G 1 = G 2 if and only if G 1 ⊆ G 2 and G 1 ⊇ G 2.

v 1

v 3 v^2 v^4

v 1

v 3 v 2

⊆

• A subgraph H of G is called an induced subgraph of G if for every two induced subgraph vertices u, v ∈ V (H) we have uv ∈ E(H) ⇔ uv ∈ E(G). In the example above H is not an induced subgraph of G. Every induced subgraph of G can be obtained by deleting vertices (and all incident edges) from G. Examples: v 1 v 3 v 2 v 4

v 1 v 3 v 2

v 1 v 2 v (^4) v 3 v 2 v 4 v 3 v 4

v 1

• Every induced subgraph of G is uniquely defined by its vertex set. We write

length δ(G) and a cycle with at least δ(G) + 1 vertices.

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

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

Proposition 6. If a graph has a closed walk with a non-repeated edge, 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 is a closed walk containing all edges of G, Eulerian tour each exactly once.

Theorem 8 (Eulerian Tour Condition, 1.8.1). A connected graph has an Eulerian tour 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 (v 1 ,... , vn) so that for every i ∈ { 1 ,... , n} the graph G

[

{v 1 ,... , vi}

]

is connected.

Operations on graphs

Definition. Let G = (V, E) and G′^ = (V ′, E′) be two graphs, U ⊆ V be a subset of 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′) and G ∩ G′^ := (V ∩ V ′, E ∩ E′). Note that G^ ∪^ G′, G^ ∩^ G′ G, G′^ ⊆ G ∪ G′^ and G ∩ G′^ ⊆ G, G′. Sometimes, we also write G + G′^ for G ∪ G′.
• G − U := G[V \ U ], G − F := (V, E \ F ) G − U , G − F , G + F

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

• For an edge e = xy in G we define G ◦ e as the graph obtained from G by G ◦ e identifying x and y and removing (if necessary) loops and multiple edges. We say that G ◦ e arises from G by contracting the edge e. contract

x

y

v 3

v 2

v 1

v 4

v 5

v 2 vxy

v 1

v 3

v 4

v 5

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

(V

2

\

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 , denoted by g(G), is the length of a shortest cycle in G. If girth,^ g(G) G is acyclic, its girth is said to be ∞.
• The circumference of G is the length of a longest cycle in G. If G is acyclic, circumference its circumference is said to be 0.
• The graph G is called Hamiltonian if G has a spanning cycle, i.e., there is a Hamiltonian cycle in G that contains every vertex of G. In other words, G is Hamiltonian if and only if its circumference is |V |.
• The graph G is called traceable if G has a spanning path, i.e., there is a traceable path in G that contains every vertex of G.
• For two vertices u and v in G, the distance between u and v , denoted by distance,^ d(u, v) d(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 ∞.
• The diameter of G diameter, diam(G)

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

diam(G) = max u,v∈V d(u, v).

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

rad(G) = min u∈V max v∈V d(u, v).

• If there is a vertex ordering v 1 ,... , vn of G and a d ∈ N such that

|N (vi) ∩ {vi+1,... , vn}| ≤ d,

for all i ∈ [n − 1] then G is called d-degenerate. The minimum d for which d-degenerate G is d-degenerate is called the degeneracy of G. degeneracy

4 Matchings

Definition.

• A matching (independent edge set) is a vertex-disjoint union of edges, i.e., matching the union of pairwise non-adjacent edges. ...
• A matching in G is a subgraph of G isomorphic to a matching. We denote the size of the largest matching in G by ν(G). ν(G)
• A vertex cover in G is a set of vertices U ⊆ V such that each edge in E vertex cover is incident to at least one vertex in U. We denote the size of the smallest vertex cover in G by τ (G). τ (G)

U

• A k-factor of G is a k-regular spanning subgraph of G. k-factor
• A 1-factor of G is also called a perfect matching since it is a matching of perfect matching largest possible size in a graph of order |V |. Clearly, G can only contain a perfect matching if |V | is even.

Theorem 15 (Hall’s Marriage Theorem, 2.1.2). Let G be a bipartite graph with partite sets 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 16 (Tutte’s Theorem, 2.2.1). For S ⊆ V define q(S) to be the number of odd components of G − S, i.e., the number of connected components of G − S consisting of an odd number of vertices. A graph G has a perfect matching if and only if q(S) ≤ |S| for all S ⊆ V.

S

odd

odd

odd

|S| ≥ odd components of G − S

Corollary 17.

• Let G be bipartite with partite sets A and B such that |N (S)| ≥ |S| − d for all S ⊆ A, and a fixed positive integer d. Then G contains a matching of size at least |A| − d.
• A k-regular bipartite graph has a perfect matching.
• A k-regular bipartite graph has a proper k-edge coloring.

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

• For all functions f : V → N ∪ { 0 } an f -factor of G is a spanning subgraph H f^ -factor of G such that degH (v) = f (v) for all v ∈ V.
• Let f : V → N ∪ { 0 } be a function with f (v) ≤ deg(v) for all v ∈ V. We can construct the auxiliary graph T (G, f ) by replacing each vertex v with vertex T^ (G, f^ ) sets 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 a complete bipartite graph between A(v) and B(v) for each vertex v.

1 v 2

1

3 v 1

2

3 2

∅

B(v 1 ) (^) A(v 1 )^ B(v 2 ) A(v 2 ) →

• Let H be a graph. An H-factor of G is a spanning subgraph of G that is H-factor a vertex-disjoint union of copies of H, i.e., a set of copies of H in G whose vertex sets form a partition of V.

5 Connectivity

Definition.

• For a natural number k ≥ 1, a graph G is called k-connected if |V (G)| ≥ k+1 k-connected and for any set U of k − 1 vertices in G the graph G − U is connected. In particular, Kn is (n − 1)-connected.
• The maximum k for which G is k-connected is called the connectivity of G , connectivity,^ κ(G) denoted by κ(G).

κ(

v 1 v 3 v^2 v^4 ) = 1, κ(Cn) = 2, κ(Kn,m) = min{m, n}.

• For a natural number k ≥ 1, a graph G is called k-linked if for any 2k distinct k-linked vertices s 1 , s 2 ,... , sk, t 1 , t 2 ,... , tk there are vertex-disjoint si-ti-paths, i = 1 ,... , k. s 1 s 2 s 3

t 1 t 2 t 3

s 1 s 2 s 3

t′ 1 = t 2 t′ 2 = t 3 t′ 3 = t 1

• For a graph G = (V, E) a set X ⊆ V ∪ E of vertices and edges of G is called a cut set of G if G − X has more connected components than G. If a cut set cut set consists of a single vertex v, then v is called a cut vertex of G; if it consists cut vertex of a single edge e, then e is called a cut edge or bridge of G. cut edge, bridge
• For a natural number ≥ 1, a graph G is called-edge-connected if G is -edge-connected non-trivial and for any set F ⊆ E of fewer than edges in G the graph G − F is connected.
• The edge-connectivity of G edge-connectivity, κ′(G)

is the maximum such that G is-edge-connected. It is denoted by κ′(G) or λ(G). G non-trivial tree ⇒ λ(G) = 1, G cycle ⇒ λ(G) = 2.

Clearly, for every k, ≥ 2, if a graph is k-connected, k-linked or-edge-connected, then it is also (k − 1)-connected, (k − 1)-linked or (` − 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 22. For any connected, non-trivial graph G we have

κ(G) ≤ λ(G) ≤ δ(G).

K 100 K 100

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 in G we say that X separates A and B if each A-B-path contains an element of X. separate

A B

v 1 v^2 v 3

u 1 u 2 u^3

e 1 e 2

e 3 e 4

e 5

Some sets separating A and B: {e 1 , e 4 , e 5 }, {e 1 , u 2 }, {u 1 , u 3 , v 3 }

Note that if X separates A and B, then necessarily A ∩ B ⊆ X.

Theorem 23 (Menger’s Theorem, 3.3.1). For any graph G and any two vertex sets A, B ⊆ V (G) we have

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

Corollary 24. 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 25 (Global Version of Menger’s Theorem, 3.3.6). A graph G is k- connected if and only if for any two vertices a, b in G there exist k independent a-b-paths.

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

Definition. For a graph G = (V, E) the line graph L(G) of G is the graph L(G) = line graph^ L(G) (E, E′), where

E′^ =

{e 1 , e 2 } ∈

E

: e 1 adjacent to e 2 in G

Gi

Theorem 27 (3.1.1). A graph is 2-connected if and only if it has an ear-decomposition.

Lemma 28. If G is 3-connected, then there exists an edge e of G such that G ◦ e is also 3-connected.

Theorem 29 (Tutte, 3.2.3). A graph G is 3-connected if and only if there exists a sequence of graphs G 0 , G 1 ,... , Gk, such that

• G 0 = K 4 ,
• for each i = 1,... , k the graph Gi has two adjacent vertices x′, x′′^ of degree at least 3, so that Gi− 1 = Gi ◦ x′x′′, and
• Gk = G.

x

x′ x′′

y y′

y′′

Definition. Let G be a graph. A maximal connected subgraph of G without a cut vertex is called a block of G. In particular, the blocks of G are exactly the block bridges and the maximal 2-connected subgraphs of G. The block-cut-vertex graph or block graph block-cut-vertex graph

of G is a bipartite graph H whose partite sets are the blocks of G and the cut vertices of G, respectively. There is an edge between a block B and a cut vertex a if and only if a ∈ B, i.e., the block contains the cut vertex.

B 1 B 2 B (^3) B 4

B 5 B 6

v 1 v (^2) v 3

v 4 v 5

B 1 B 2

v 1 v 2

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

Theorem 30. The block-cut-vertex graph of a connected graph is a tree.

6 Planar graphs

This section deals with graph drawings. We restrict ourselves to graph drawings in the plane R^2. It is also feasible to consider graph drawings in other topological spaces, such as the torus.

Definition.

• The straight line segment straight line segment

between p ∈ R^2 and q ∈ R^2 is the set {p+λ(q−p) : 0 ≤ λ ≤ 1 }.

• A homeomorphism is a continuous function that has a continuous inverse homeomorphism function.
• Two sets A ∈ R^2 and B ∈ R^2 are said to be homeomorphic if there is a homeomorphic homeomorphism f : A → B.
• A polygon is a union of finitely many line segments that is homeomorphic polygon to the circle S^1 := {x ∈ R^2 : ‖x‖ = 1}.
• An arc is a subset of R^2 which is the union of finitely many straight line arc segments and is homeomorphic to the closed unit interval [0, 1]. The images of 0 and 1 under such a homeomorphism are the endpoints of the arc. If P endpoint of arc is an arc with endpoints p and q, then P links them and runs between them. The set P \ {p, q} is the interior of P , denoted by P˚. interior of arc
• Let O ⊆ R^2 be an open set. Being linked by an arc in O is an equivalence relation on O. The corresponding equivalence classes are the regions of O. region A closed set X ⊆ R^2 is said to separate O if O \ X has more regions than separate O. The frontier of a set X ⊆ R^2 is the set Y of all points y ∈ R^2 such frontier that every neighbourhood of y meets both X and R^2 \ X. Note that if X is closed, its frontier lies in X, while if X is open, its frontier lies in R^2 \ X.
• A plane graph is a pair (V, E) of finite sets with the following properties plane graph (the elements of V are again called vertices, those in E edges): 1. V ⊆ R^2 ; 2. every e ∈ E is an arc between two vertices; 3. different edges have different sets of endpoints; 4. the interior of an edge contains no vertex and no point of any other edge.

A plane graph (V, E) defines a graph G on V in a natural way. As long as no confusion can arise, we shall use the name G of this abstract graph also