GRAPH THEORY - PART 1 doc

Introduction Graph theory can be said to have its beginning in 1736 when EULER considered the general case of the Königsberg bridge problem: Is there a walk-ing route that crosses each

Lecture Notes on

GRAPH THEORY

Tero Harju

Department of Mathematics

University of Turku

FIN-20014 Turku, Finland

e-mail: harju@utu.fi

2007

1 Introduction : : : : : : : : : : : : : : : : : : : : : : : : : : : : 2 1.1 Graphs and their plane figures 4 1.2 Subgraphs 7 1.3 Paths and cycles 11

2 Connectivity of Graphs: : : : : : : : : : : : : : : : : : : : : : : : 16 2.1 Bipartite graphs and trees 16 2.2 Connectivity 24

3 Tours and Matchings: : : : : : : : : : : : : : : : : : : : : : : : : 30 3.1 Eulerian graphs 30 3.2 Hamiltonian graphs 32 3.3 Matchings 36

4 Colourings : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 43 4.1 Edge colourings 43 4.2 Ramsey Theory 47 4.3 Vertex colourings 52

5 Graphs on Surfaces : : : : : : : : : : : : : : : : : : : : : : : : : 60 5.1 Planar graphs 60 5.2 Colouring planar graphs 67 5.3 Genus of a graph 74

6 Directed Graphs: : : : : : : : : : : : : : : : : : : : : : : : : : : 83 6.1 Digraphs 83 6.2 Network Flows 89

Index: : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 96

Introduction

Graph theory can be said to have its beginning in

1736 when EULER considered the (general case of

the) Königsberg bridge problem: Is there a

walk-ing route that crosses each of the seven bridges

of Königsberg exactly once? (Solutio

Problema-tis ad geometriam situs pertinenProblema-tis, Commentarii

Academiae Scientiarum Imperialis Petropolitanae 8

(1736), pp 128-140.)

It took 200 years before the first book on graph theory was written This was done by

KÖNIGin 1936 (“Theorie der endlichen und unendlichen Graphen”, Teubner, Leipzig, 1936 Translation in English, 1990.) Since then graph theory has developed into an extensive and popular branch of mathematics, which has been applied to many problems in mathematics, computer science, and other scientific and not-so-scientific areas For the history of early graph theory, see

N.L BIGGS, R.J LLOYD AND R.J WILSON, “Graph Theory 1736 – 1936”, Clarendon Press, 1986

There seem to be no standard notations or even definitions for graph theoretical objects This is natural, because the names one uses for these objects reflect the applications So, for instance, if we consider a communications network (say, for email) as a graph, then the computers, which take part in this network, are called nodes rather than vertices or points

On the other hand, other names are used for molecular structures in chemistry, flow charts in programming, human relations in social sciences, and so on

These lectures study finite graphs and majority of the topics is included in

J.A BONDY ANDU.S.R MURTY, “Graph Theory with Applications”, Macmillan, 1978

R DIESTEL, “Graph Theory”, Springer-Verlag, 1997

F HARARY, “Graph Theory”, Addison-Wesley, 1969

D.B WEST, “Introduction to Graph Theory”, Prentice Hall, 1996

R.J WILSON, “Introduction to Graph Theory”, Longman, (3rd ed.) 1985

In these lectures we study combinatorial aspects of graphs For more algebraic topics and

methods, see

N BIGGS, “Algebraic Graph Theory”, Cambridge University Press, (2nd ed.) 1993

and for computational aspects, see

S EVEN, “Graph Algorithms”, Computer Science Press, 1979

In these lecture notes we mention several open problems that have gained respect among the researchers Indeed, graph theory has the advantage that it contains easily formulated open problems that can be stated early in the theory Finding a solution to any one of these problems

is on another layer of difficulty

Sections with a star () in their heading are optional

Notations and notions

For a finite setX,jXjdenotes its size (cardinality, the number of its elements)

Let

[1; n℄ = f1; 2; : : ng;

and in general,

[i; n℄ = fi; i + 1; : : ng for integersi  n

For a real numberx, the floor and the ceiling ofxare the integers

= max fk 2 Z j k  xg and dxe = minfk 2 Z j x  kg:

A familyfX

1

; X 2

; : ; X k

gof subsetsX

i

 Xof a setXis a partition ofX, if

X = [

i2[1;k℄

X

i and X

i

\ X j

= ;for all differentiandj :

For two setsXandY,

X  Y = f(x; y) j x 2 X; y 2 Y g

is their Cartesian product.

For two setsXandY,

X4Y = (X n Y ) [ (Y n X)

is their symmetric difference HereX n Y = fx j x 2 X; x 2 = Y g

 Two numbers n; k 2 N (often n = jX jand k = jY jfor sets X and Y) have the same

parity, if both are even, or both are odd, that is, if n  k (mod 2) Otherwise, they have opposite parity

Graph theory has abundant examples of NP-complete problems Intuitively, a problem is

in P1 if there is an efficient (practical) algorithm to find a solution to it On the other hand,

a problem is in NP2, if it is first efficient to guess a solution and then efficient to check that this solution is correct It is conjectured (and not known) that P 6= NP This is one of the great problems in modern mathematics and theoretical computer science If the guessing in NP-problems can be replaced by an efficient systematic search for a solution, then P=NP For any one NP-complete problem, if it is in P, then necessarily P=NP

1

Solvable – by an algorithm – in polynomially many steps on the size of the problem instances.

Solvable nondeterministically in polynomially many steps on the size of the problem instances.

1.1 Graphs and their plane figures 4

1.1 Graphs and their plane figures

LetV be a finite set, and denote by

E(V ) = ffu; v j u; v 2 V; u 6= v :

the subsets ofV of two distinct elements

DEFINITION A pairG = (V; E)with E  E(V ) is called a graph (onV) The elements

ofV are the vertices, and those ofE the edges of the graph The vertex set of a graphGis denoted byV and its edge set byE

G ThereforeG = (V

G

; E G )

In literature, graphs are also called simple graphs; vertices are called nodes or points; edges are called lines or links The list of alternatives is long (but still finite).

A pair fu; v is usually written simply as uv Notice that then uv = vu In order to simplify notations, we also writev 2 Ginstead ofv 2 V

DEFINITION For a graphG, we denote

 G

= jV G

j and "

G

= jE G j

The number

Gof the vertices is called the order ofG, and"

Gis the size ofG For an edge

e = uv 2 E

G, the verticesuandvare its ends Verticesuandvare adjacent or neighbours,

ife = uv 2 E

G Two edgese

1

= uvande

2

= uwhaving a common end, are adjacent with

each other

A graphGcan be represented as a plane figure by drawing

a line (or a curve) between the points uandv (representing

vertices) ife = uvis an edge ofG The figure on the right is

a drawing of the graph GwithV

G

= fv 1

; v 2

; v 3

; v 4

; v 5

; v 6 g andE

G

= fv

1

v

2

; v 1 v 3

; v 2 v 3

; v 2 v 4

; v 5 v 6

g

v 1

v 2

v 3

v 4 v 5

v 6

Often we shall omit the identities (namesv) of the vertices in our figures, in which case the vertices are drawn as anonymous circles

Graphs can be generalized by allowing loopsv and parallel (or multiple) edges between vertices to obtain a multigraph G = (V; E; ), where E = fe

1

; e 2

; : ; e m

g is a set (of symbols), and : E ! E(V ) [ fvv j v 2 V gis a function that attaches an unordered pair of vertices to eache 2 E: (e) = uv

Note that we can have (e

1

= (e

2 This is drawn in the figure ofGby placing two (parallel) edges that connect the

common ends On the right there is (a drawing of) a

multi-graphGwith verticesV = fa; b; and edges (e

1

= aa, (e

2

= ab, (e

3

= , and (e

4

=

a b

1.1 Graphs and their plane figures 5

Later we concentrate on (simple) graphs.

DEFINITION We also study directed graphs or digraphs

D = (V; E), where the edges have a direction, that is, the

edges are ordered:E  V  V In this case,uv 6= vu

The directed graphs have representations, where the edges are drawn as arrows A digraph can contain edgesuvandvuof opposite directions.

Graphs and digraphs can also be coloured, labelled, and weighted:

DEFINITION A function : V

G

! K is a vertex colouring ofGby a setK of colours A function : E

G

! K is an edge colouring ofG Usually,K = [1; k℄for somek  1

IfK  R (oftenK  N), thenis a weight function or a distance function.

Isomorphism of graphs

DEFINITION Two graphs Gand H are isomorphic, denoted by G



H, if there exists a bijection : V

G

! V such that

uv 2 E G ( ) (u)(v) 2 E

H

for allu; v 2 G

HenceG and H are isomorphic if the vertices of H are renamings of those of G Two

isomorphic graphs enjoy the same graph theoretical properties, and they are often identified.

In particular, all isomorphic graphs have the same plane figures (excepting the identities of the vertices) This shows in the figures, where we tend to replace the vertices by small circles, and talk of ‘the graph’ although there are, in fact, infinitely many of such graphs

Example 1.1 The following graphs are

iso-morphic Indeed, the required isomorphism

is given by v

1

7! 1, v

2 7! 3, v

3 7! 4, v

4

7! 2,v

5

1

v 2

v 3

v 4

v

3

4 2

5

Isomorphism Problem Does there exist an efficient algorithm to check whether any two

given graphs are isomorphic or not?

The following table lists the number2

( n 2 )

of graphs on a given set of nvertices, and the number of nonisomorphic graphs onnvertices It tells that at least for computational purposes

an efficient algorithm for checking whether two graphs are isomorphic or not would be greatly appreciated

1.1 Graphs and their plane figures 6

graphs 1 2 8 64 1024 32 768 2 097 152 268 435 456 2

36

> 6
10

10 nonisomorphic 1 2 4 11 34 156 1044 12 346 274 668

Other representations

Plane figures catch graphs for our eyes, but if a problem on graphs is to be programmed, then

these figures are (to say the least) unsuitable Matrices of integers are ideal for computers, since every respectable programming language has array structures for these, and computers are good in crunching numbers

LetV = fv

1

; : ; v gbe ordered The adjacency matrix

ofGis then n-matrixMwith entriesM

ij

= 1orM

ij

=

0 according to whether v

i v j

2 E

G or not For instance, the graphs of Example 1.1 has an adjacency matrix on the

right Notice that the adjacency matrix is always symmetric

(with respect to its diagonal consisting of zeros)

0

B B B



0 1 1 0 1

1 0 0 1 1

1 0 0 1 0

0 1 1 0 0

1 1 0 0 0

1

C C C A

A graph has usually many different adjacency matrices, one for each ordering of its setV

of vertices The following result is obvious from the definitions

Theorem 1.1 Two graphs Gand H are isomorphic if and only if they have a common ad-jacency matrix Moreover, two isomorphic graphs have exactly the same set of adad-jacency matrices.

Graphs can also be represented by sets For this, letX = fX

1

; X 2

; : ; X

n be a fam-ily of subsets of a set X, and define the intersection graphG

X as the graph with vertices X

1

; : ; X

n, and edgesX

i X

jfor alliandj(i 6= j) withX

i

\ X j 6= ;

Theorem 1.2 Every graph is an intersection graph of some family of subsets.

Proof LetGbe a graph, and define, for allv 2 G, a set

X v

= ffv; ug j vu 2 E

G g:

ThenX

u

\ X

v

6= ;if and only ifuv 2 E

Lets(G)be the smallest size of a base setXsuch thatGcan be represented as an inter-section graph of a family of subsets ofX, that is,

s(G) = minfjXj j G

 G

Xfor someX  2

X

g :

How small cans(G)be compared to the order

G(or the size"

G) of the graph? It was shown

by KOU, STOCKMEYER ANDWONG (1976) that it is algorithmically difficult to determine the number – the problem is NP-complete

1.2 Subgraphs 7 Example 1.2 As yet another example, let A  N be a finite set of natural numbers, and let G

A

= (A; E) be the graph defined onV

G A

= A such thatrs 2 E (= E

G A )if and only if

r ands(forr 6= s) have a common divisor> 1 As an exercise, we state: All graphs can be

represented in the formG

Afor some setAof natural numbers.

1.2 Subgraphs

Ideally, in a problem the local properties of a graph determine a solution In such a situation

we deal with (small) parts of the graph (subgraphs), and a solution can be found to the problem

by combining the information determined by the parts For instance, as we shall see later on, the existence of an Euler tour is very local, it depends only on the number of the neighbours

of the vertices

Degrees of vertices

DEFINITION Letv 2 Gbe a vertex a graphG The neighbourhood ofvis the set

N G (v) = fu 2 G j vu 2 E

G

g :

The degree ofvis the number of its neighbours:

d G (v) = jN

G (v)j :

Ifd

G

(v) = 0, thenvis said to be isolated inG, and ifd

G (v) = 1, thenvis a leaf of the graph The minimum degree and the maximum degree ofGare defined as

Æ(G) = minfd

G (v) j v 2 Gg and
(G) = maxfd

G (v) j v 2 Gg :

The following lemma, due to EULER(1736), tells that if several people shake hands, then the number of hands shaken is even

Lemma 1.1 (Handshaking lemma) For each graphG,

X

v2G d G (v) = 2
"

G :

Moreover, the number of vertices of odd degree is even.

Proof Every edge e 2 E

G has two ends The second claim follows immediately from the

Lemma 1.1 holds equally well for multigraphs, whend

G (v) is defined as the number of edges that havevas an end, and when a loopv is counted twice.

Note that the degrees of a graphG do not determine G Indeed, there are graphs G = (V; E

G

)andH = (V; E

H )on the same set of vertices that are not isomorphic, but for which

for all

1.2 Subgraphs 8

DEFINITION Let G be a graph A 2-switch

(u; v; x; y) ofG, foruv; xy 2 E

Gand ux; v 2 = E

G, replaces the edgesuvandxybyuxandv u

v

x y

u v

x y

Before proving Berge’s switching theorem we need the following tool

Lemma 1.2 LetGbe a graph of ordernwith a degree sequenced

1

 d 2



 d , where

d

G

(v

i

) = d

i There is a graphG

0

which is obtained fromGby a sequence of2-switches such thatN

G

0 (v

1

= fv

2

; : ; v d 1 +1

g.

Proof Denoted =
(G) (= d

1 Suppose that there exists a vertexv

iwith2  i  d + 1 such thatv

1

v

i

=

2 E

G Sinced

G (v 1

= d, there exists av

j withj  d + 2such thatv

1 v j

2 E

G Hered

i

 d

j, since

j > i Sincev

1

v j

2 E

G, there exists a v

t (2  t  n) such thatv

i

v

t

2 E

G, butv

j v t

=

2 E

G We can now perform

a 2-switch with respect to the vertices v

1

; v j v i

; v

t This gives a new graphH, where v

1 v i

2 E

H andv

1 v j

=

2 E

H, and the other neighbours ofv

1remain to be its neighbours

v 1

v i

v j

v t

When we repeat this process for all indicesiwithv

1 v i

=

2 E

Gfor2  i  d + 1, we obtain

a graphG

0

Theorem 1.3 (BERGE (1973)) Two graphs G and H on a common vertex set V satisfy

d

G

(v) = d

H

(v) for allv 2 V if and only ifH can be obtained fromG by a sequence of

2-switches.

Proof If a graphH is obtained fromGby a2-switch, then clearlyH has the same degrees

asG

In the other direction, we use induction on the order 

G Let G and H have the same degrees, and letd =
(G) By Lemma 1.2, there are sequences of2-switches that transform

G to G

0

and H to H

0 such that N

G 0(v 1

= fv 2

; : ; v d+1

g = N

H 0 (v

1 Now the graphs G

0

v

1 and H

0

v

1 have the same degrees By induction hypothesis, G

0 , and thus also G, can be transformed to H

0

by a sequence of2-switches Finally, we observe that H

0 can be transformed toHby the ‘inverse sequence’ of2-switches, and this proves the claim u

DEFINITION Letd

1

; d 2

; : ; d be a descending sequence of nonnegative integers, that is, d

1

 d

2



 d Such a sequence is said to be graphical, if there exists a graph G = (V; E)withV = fv

1

; v 2

; : ; v gsuch thatd

i

= d G (v i )for alli Using the next result recursively one can decide whether a sequence of integers is graphical

or not

1.2 Subgraphs 9

Theorem 1.4 (HAVEL (1955), HAKIMI(1962)) A sequenced

1

; d 2

; : ; d (withd

1

 1and

n  2) is graphical if and only if

d 2 1; d 3 1; : : d

d 1 +1 1; d d 1 +2

; d d 1 +3

is graphical (when put into nonincreasing order).

Proof (() ConsiderGof ordern 1with vertices (and degrees)

d G (v 2

= d 2 1; : : d

G (v d 1 +1 ) = d d 1 +1 1;

d G (v d 1 +2 ) = d d 1 +2

; : ; d G (v n ) = d

as in (1.1) Add a new vertex v

1 and the edgesv

1 v

i for alli 2 [2; d

d1 +1

℄ Then in this new graphH,d

H

(v

1

= d

1, andd

H (v i ) = d

ifor alli ()) Assume d

G (v i ) = d

i By Lemma 1.2 and Theorem 1.3, we can suppose that N

G

(v

1

= fv

2

; : ; v

d 1 +1

g But now the degree sequence ofG v

1is in (1.1) u

Example 1.3 Consider the sequences = 4; 4; 4; 3; 2; 1 By Theorem 1.4,

sis graphical ( ) 3; 3; 2; 1; 1is graphical

2; 1; 1; 0is graphical 0; 0; 0is graphical:

The last sequence corresponds to a discrete graph K

3, and hence also our original sequence sis graphical Indeed, the

graphGon the right has this degree sequence

v1

v 2

v3

v 4

v5 v6

Special graphs

DEFINITION A graphG = (V; E)is trivial, if it has only one vertex, i.e.,

G

= 1; otherwise

Gis nontrivial.

The graph G = K

V is the complete graph onV, if every two vertices are adjacent: E = E(V ) All complete graphs

of ordernare isomorphic with each other, and they will be

denoted byK

n

K 6

The complement ofGis the graph GonV

G, whereE

G

= fe 2 E(V ) j e 2 = E

G

g The complementsG = K

V of the complete graphs are called discrete graphs In a discrete graph

E

G

= ; Clearly, all discrete graphs of ordernare isomorphic with each other

A graphGis said to be regular, if every vertex ofGhas the same degree If this degree is equal tor, thenGisr-regular or regular of degreer