GRAPH THEORY - PART 4 pptx

4.1 Edge colourings Colourings of edges and vertices of a graphGare useful, when one is interested in classifying relations between objects.. In the chromatic theory,Gis first given and

4.1 Edge colourings

Colourings of edges and vertices of a graphGare useful, when one is interested in classifying relations between objects

There are two sides of colourings In the general case, a graphG with a colouring is given, and we study the properties of this pair G

= (G; ) This is the situation, e.g., in transportation networks with bus and train links, where the colour (buss, train) of an edge

tells the nature of a link

In the chromatic theory,Gis first given and then we search for a colouring that the satisfies required properties One of the important properties of colourings is ‘properness’ In a proper colouring adjacent edges or vertices are coloured differently

Edge chromatic number

DEFINITION Ak-edge colouring : E

G

! [1; k℄of a graphGis an assignment ofkcolours

to its edges We writeG

to indicate thatGhas the edge colouring

A vertexv 2 Gand a colouri 2 [1; k℄are incident with each other, if(vu) = ifor some

vu 2 E

G Ifv 2 Gis not incident with a colouri, theniis available forv

The colouringis proper, if no two adjacent edges obtain the same colour:(e

1 6= (e 2 for adjacente

1ande 2.

The edge chromatic number

0 (G)ofGis defined as

0 (G) = minfk j there exists a properk-edge colouring ofGg :

Ak-edge colouring  can be thought of as a partition fE

1

; E 2

; : ; E k

g ofE

G, where E

i

= fe j (e) = ig Note that it is possible thatE

i

= ;for somei We adopt a simplified notation

G [i 1

; i ; : ; i

t

= G[E i 1 [ E i 2 [

[ E

i

℄ for the subgraph ofGconsisting of those edges that have a colouri ,i , , ori

t That is, the edges having other colours are removed

i in a properk-edge colouring is a matching Moreover, for each graphG,
(G)  

0 (G)  "

G.

4.1 Edge colourings 44 Example 4.1 The three numbers in Lemma 4.1 can be equal This happens, for instance,

whenG = K

1;nis a star But often the inequalities are strict

A star, and a graph with

0 (G) = 4

Optimal colourings

We show that for bipartite graphs the lower bound is always optimal:

0 (G) =
(G)

colouring (that need not be proper), in which both colours are incident with each vertexvwith

d

G

(v)  2.

(1) Suppose first thatGis eulerian IfGis an even cycle, then a 2-edge colouring exists

as required Otherwise, since nowd

G (v)is even for allv,Ghas a vertexv

1withd

G (v 1

 4 Lete

1

e

2

: : e

tbe an Euler tour ofG, wheree

i

= v i v i+1(andv

t+1

= v

1) Define

(e i ) = ( 1; ifiis odd; 2; ifiis even:

Hence the ends of the edgese

ifori 2 [2; t 1℄are incident with both colours All vertices are among these ends The conditiond

G (v 1

 4guarantees this forv

1 Hence the claim holds in the eulerian case

(2) Suppose then thatGis not eulerian We define a new graphG

0by adding a vertexv

0

toGand connectingv

0to eachv 2 Gof odd degree

In G

0 every vertex has even degree including v

0 (by the handshaking lemma), and hence G

0 is eulerian Let e

0

e

1

: : e

t be an eulerian tour of G

0, where e

i

= v i v i+1

By the previous case, there is a required colouring ofG

0

as above Now,  restricted to E

G is a colouring of G as required by the claim, since each vertexv

i with odd degree d

G

(v

i

)  3is entered and departed at least once in the tour

by an edge of the original graphG:e

i 1 e

i

v 0 1

2 1 2 1 2

u

DEFINITION For ak-edge colouringofG, let

(v) = jfi j vis incident withi 2 [1; k℄gj :

A -edge colouring is an improvement of , if

X v2G (v) >

X v2G (v) :

Also,is optimal, if it cannot be improved.

Notice that we always have 

(v)  d

G (v), and ifis proper, then 

(v) = d

G (v), and

in this case is optimal Thus an improvement of a colouring is a change towards a proper colouring Note also that a graphGalways has an optimalk-edge colouring, but it need not have any properk-edge colourings

The next lemma is obvious

(v) = d

G (v)for all vertices

v 2 G.

colouriis available forv, and the colourjis incident withvat least twice Then the connected componentHofG

[i; j℄that containsv, is an odd cycle.

2-edge colouring : E

H

! fi; jg, in which bothiandjare incident with each vertexxwith d

H

(x)  2 (We have renamed the colours1and2toiandj.) We obtain a recolouring of

Gas follows:

(e) =

( (e); ife 2 E

H

;

(e); ife 2 = E

H : Since d

H

(v)  2 (by the assumption on the colour j) and in  both colours iand j are now incident with v, 

(v) =

(v) + 1 Furthermore, by the construction of , we have 

(u) 

(u)for allu 6= v Therefore

P u2G (u) >

P u2G (u), which contradicts the

0 (G) =
(G).

were av 2 Gwith 

(v) < d

G (v), then by Lemma 4.4,Gwould contain an odd cycle But a bipartite graph does not contain such cycles Therefore, for all verticesv, 

(v) = d

G (v) By Lemma 4.3,is a proper colouring, and
= 

0

Vizing’s theorem

In general we can have 

0 (G) >
(G) as one of our examples did show The following important theorem, due to VIZING, shows that the edge chromatic number of a graphGmisses

(G)by at most one colour

0 (G) 
(G) + 1.

0 (G) 
+ 1 Suppose on the contrary that , and let be an optimal -edge colouring of

4.1 Edge colourings 46

We have (trivially)d

G (u) <
+ 1 < 

0 (G)for allu 2 G, and so

Moreover, by the counter hypothesis,is not a proper colouring, and hence there exists a

v 2 Gwith 

(v) < d

G (v), and hence a colouri that is incident withvat least twice, say

(vu 1

1

; u 2

; : such that

(vu j ) = i

j and i

j+1

= b(u j ) :

Indeed, let u

1 be as in (4.1) Assume we have already found the vertices u

1

; : ; u

j, with

j  1, such that the claim holds for these Suppose, contrary to the claim, that v is not incident withb(u

j ) = i j+1

We can recolour the edges vu

` by i

`+1 for ` 2 [1; j℄, and obtain in this way an improvement of Herevgains a new

colouri j+1 Also, each u

`gains a new colouri

`+1 (and may loose the colour i

`) Therefore, for each u

` either its num-ber of colours remains the same or it increases by one This

contradicts the optimality of, and proves Claim 2

Now, let tbe the smallest index such that for some r < t,

i

t+1

= i Such an indextexists, becaused

G (v)is finite

x u1 u2

ur

.

..

u

r 1

i1 i1 i 2

i

r 1

ir = it+1 i

Let  be a recolouring ofGsuch that for 1  j  r 1,

(vu

j

) = i j+1, and for all other edges e, (e) = (e)

Indeed, 

(v) =

(v) and 

(u) 

(u)for allu, since eachu

j (1  j  r 1) gains a new colourj

i+1although it may loose one of its old colours

x u1 u2

ur

.

..

u

r 1

i1 i 2 i3

i r

ir = it+1 i

Let then the colouring be obtained from by recolouring

the edgesvu

jbyi

j+1forr  j  t Now,vu

tis recoloured

byi = i

t+1

Indeed, the fact i = i

t+1 ensures that i is a new colour incident withu

t, and thus that (u

t )  (u t ) For all other vertices, follows as for

u1 u2

ur

v

ut

.

..

ur 1

i 1 i 2 i3

ir i r+1 i r

By Claim 1, there is a colouri = b(v)that is available forv By Lemma 4.4, the connected componentsH

1 ofG

[i 0

; i ℄and H

2ofG [i

0

; i ℄containing the vertexv are cycles, that is, H

1is a cycle(vu

r 1 )P 1 (u r v)andH

2is a cycle(vu

r 1 )P 2 (u t v), where bothP

1 : u

r 1

?

! u r andP

2

: u

r 1

?

! u

tare paths However, the edges ofP

1andP

2have the same colours with respect toand (eitheri ori ) This is not possible, sinceP

1ends inu

rwhileP

2ends in

a different vertexu

0 (G) = 4for the Petersen graph Indeed, by Vizing’ theorem,

0

(G) = 3or4 Suppose3colours suffice LetC : v

1

! : : ! v

5

! v

1be the outer cycle andC

0

: u

1

! : : ! u

5

! u

1the inner cycle ofGsuch thatv

i u i

2 E

Gfor alli Observe that every vertex is adjacent to all colours1; 2; 3 NowCuses one colour (say1) once and the other two twice This can be done uniquely (up to permutations):

v 1 1

! v 2 2

! v 3 3

! v 4 2

! v 5 3

! v 1 :

Hencev

1

2

! u 1, v

2 3

! u

2,v 3 1

! u 3, v

4 1

! u 4, v

5 1

! u 5 However, this means that 1cannot

be a colour of any edge inC

0 SinceC

0 needs three colours, the claim follows

Edge Colouring Problem Vizing’s theorem (nor its present proof) does not offer any

char-acterization for the graphs, for which 

0 (G) =
(G) + 1 In fact, it is one of the famous open problems of graph theory to find such a characterization The answer is known (only) for some special classes of graphs By HOLYER(1981), the problem whether

0 (G)is
(G)

or
(G) + 1is NP-complete

The proof of Vizing’s theorem can be used to obtain a proper colouring ofGwith at most

(G) + 1colours, when the word ‘optimal’ is forgotten: colour first the edges as well as you can (if nothing better, then arbitrarily in two colours), and use the proof iteratively to improve the colouring until no improvement is possible – then the proof says that the result is a proper colouring

4.2 Ramsey Theory

In general, Ramsey theory studies unavoidable patterns in combinatorics We consider an instance of this theory mainly for edge colourings (that need not be proper) A typical example

of a Ramsey property is the following: given 6 persons each pair of whom are either friends

or enemies, there are then 3 persons who are mutual friends or mutual enemies In graph theoretic terms this means that each colouring of the edges ofK

6with 2 colours results in a monochromatic triangle

Turan’s theorem for complete graphs

We shall first consider the problem of finding a general condition forK

p to appear in a graph

It is clear that every graph contains , and that every nondiscrete graph contains 2.

4.2 Ramsey Theory 48

DEFINITION A completep-partite graphG

consists ofpdiscrete and disjoint induced

sub-graphs G

1

; G

2

; : ; G

p

 G, whereuv 2 E

G

if and only ifuandvbelong to different parts,

G

iandG

j withi 6= j

Note that a complete p-partite graph is

com-pletely determined by its discrete partsG

i,i 2 [1; p℄

Letp  3, and letH = H

n;pbe the complete(p 1)-partite graph of ordern = t(p 1)+r, where r 2 [1; p 1℄and t  0, such that there arer parts H

1

; : ; H

r of order t + 1and

p 1 rpartsH

r+1

; : ; H

p 1 of ordert(whent > 0) (Hereris the positive residue ofn modulo(p 1), and is thus determined bynandp.)

By its definition,K

p

* H One can compute that the number"

H of edges ofHis equal to

T (n; p) =

2(p 1)

n 2 r 2

 1 r



The next result shows that the above boundT (n; p)is optimal

G

> T (n; p) edges, then G

contains a complete subgraphK

p.

ont Ift = 0, thenT (n; p) = n(n 1)=2, and there is nothing to prove

Suppose then thatt  1, and letGbe a graph of ordernsuch that"

Gis maximum subject

to the conditionK

p

* G Now G contains a complete subgraph G[A℄ = K

p 1, since adding any one edge to G results in aK

p, andp 1vertices of thisK

pinduce a subgraphK

p 1

 G Eachv 2 = A is adjacent to at most p 2 vertices of A; otherwise G[A [ fvg℄ = K p. Furthermore,K

p

* G A, and

G A

= n p + 1 Becausen p + 1 = (t 1)(p 1) + r,

we can apply the induction hypothesis to obtain"

G A

 T (n p + 1; p) Now

"

G

 T (n p + 1; p) + (n p + 1)(p 2) +

(p 1)(p 2) 2

= T (n; p) ;

When Theorem 4.3 is applied to trianglesK

3, we have the following interesting case

G

>

1 4

 2 G

edges, then Gcontains a triangleK

3.

Ramsey’s theorem

DEFINITION Let  be an edge colouring of G A subgraph H  G is said to be (i-)

The following theorem is one of the jewels of combinatorics

integerR (p; q)such that for alln  R (p; q), any 2-edge colouring ofK

n

! [1; 2℄contains a

1-monochromaticK

por a2-monochromaticK

q.

Before proving this, we give an equivalent statement Recall that a subset X  V is stable, ifG[X℄is a discrete graph

that for alln  R (p; q), any graphGof orderncontains a complete subgraph of orderpor

a stable set of orderq.

Be patient, this will follow from Theorem 4.6 The numberR (p; q)is known as the

It is clear thatR (p; 2) = pandR (2; q) = q

Theorems 4.4 and 4.5 follow from the next result which shows (inductively) that an upper bound exists for the Ramsey numbersR (p; q)

p; q  2, and

R (p; q)  R (p; q 1) + R (p 1; q) :

thus exists forp + q  5

It is now sufficient to show that ifGis a graph of orderR (p; q 1) + R (p 1; q), then it has a complete subgraph of orderpor a stable subset of orderq

Let v 2 G, and denote by A = V

G

n (N G (v) [ fvg) the set of vertices that are not adjacent tov SinceG hasR (p; q 1) + R (p 1; q) 1 vertices different from v, either jN

G

(v)j  R (p 1; q)orjAj  R (p; q 1)(or both)

Assume first thatjN

G (v)j  R (p 1; q) By the definition of Ramsey numbers,G[N

G (v)℄ contains a complete subgraphBof orderp 1or a stable subsetSof orderq In the first case,

B [ fvginduces a complete subgraphK

pinG, and in the second case the same stable set of orderqis good forG

IfjAj  R (p; q 1), thenG[A℄contains a complete subgraph of orderpor a stable subset

S of orderq 1 In the first case, the same complete subgraph of orderpis good forG, and

in the second case,S [ fvgis a stable subset ofGofqvertices This proves the claim u

A concrete upper bound is given in the following result

4.2 Ramsey Theory 50

R (p; q) 



p + q 2

 :

statement Assume thatp; q  3 By Theorem 4.6 and the induction hypothesis,

R (p; q)  R (p; q 1) + R (p 1; q)





p + q 3

 +



p + q 3



=



p + q 2



;

In the table below we give some known values and estimates for the Ramsey numbers

R (p; q) As can be read from the table1, not so much is known about these numbers

4 9 18 25 35-41 49-61 55-84 69-115 80-149

5 14 25 43-49 58-87 80-143 95-216 121-316 141-442

The first unknownR (p; p)(where p = q) is for p = 5 It has been verified that 43 

R (5; 5)  49, but to determine the exact value is an open problem

Generalizations

Theorem 4.4 can be generalized as follows

i

 2 be integers for i 2 [1; k℄with k  2 Then there exists an inte-ger R = R (q

1

; q 2

; : ; q ) such that for all n  R, anyk-edge colouring of K

n has an

i-monochromaticK

q

ifor somei.

we show thatR (q

1

; : ; q )  R (q

1

; : ; q

2

; p), wherep = R (q

k 1

; q ) Let n = R (q

1

; : ; q

2

; p), and let : E

Kn

! [1; k℄ be an edge colouring Let : E

K

n

! [1; k 1℄be obtained fromby identifying the coloursk 1andk:

(e) =

( (e) if(e) < k 1 ;

k 1 if(e) = k 1ork :

By the induction hypothesis,K

nhas ani-monochromaticK

q i for some1  i  k 2(and we are done, since this subgraph is monochromatic inK

n) orK

n has a(k 1)-monochromatic subgraph H

= K p In the latter case, by Theorem 4.4, H

 and thusK

n has a (k 1) -monochromatic or ak-monochromatic subgraph, and this proves the claim u S.P R , Small Ramsey numbers, Electronic J of Combin., 2000 on the Web

Since for each graphH,H  K

m form = 

H, we have

1

; H 2

; : ; H

kbe arbitrary graphs Then there exists an inte-gerR (H

1

; H

2

; : ; H

k )such that for all complete graphsK

n withn  R (H

1

; H 2

; : ; H k )

and for all k-edge colourings  ofK

n,K

n contains an i-monochromatic subgraph H

i for somei.

This generalization is trivial from Theorem 4.8 However, the generalized Ramsey num-bersR (H

1

; H

2

; : ; H k )can be much smaller than their counter parts (for complete graphs)

in Theorem 4.8

R (T ; K

n ) = (m 1)(n 1) + 1 ;

that is, any graphGof order at leastR (T ; K

n )contains a subgraph isomorphic toT, or the complement ofGcontains a complete subgraphK n.

Examples of Ramsey numbers

Some exact values are known in Corollary 4.2, even in more general cases, for some dear graphs (see RADZISZOWSKI’s survey) Below we list some of these results for cases, where the graphs are equal To this end, let

R k (G) = R (G; G; : : G) (ktimesG):

The best known lower bound of R

2 (G)for connected graphs was obtained by BURR AND

ERDÖS(1976),

R 2 (G) 

 4

G 1 3

 (Gconnected):

Here is a list of some special cases:

R 2 (P n ) = n +

j n 2 k 1;

R 2 (C n ) =

8

>

>

2n 1 ifn  5andnodd; 3n=2 1 ifn  6andneven;

R 2 (K 1;n ) = ( 2n 1 ifnis even; 2n ifnis odd;

R 2 (K 2;3

2 (K 3;3 ) = 18:

The valuesR

2

(K

2;n )are known forn  16, and in general,R

2 (K 2;n )  4n 2 The value R

2

(K

2;17

)is either65or66

LetW

ndenote the wheel onnvertices It is a cycleC

n 1, where a vertex v with degree

is attached Note that 4 Then and

4.3 Vertex colourings 52

For three colours, much less is known In fact, the only nontrivial result for complete graphs is:R

3

(K

3

= 17 Also,128  R

3 (K 4

 235, and385  R

3 (K

5 , but no nontrivial upper bound is known forR

3 (K

5 For the squareC

4, we know thatR

3 (C 4

= 11 Needless to say that no exact values are known forR

k (K n )fork  4andn  3

It follows from Theorem 4.4 that for any completeK n, there exists a graph G(well, any sufficiently large complete graph) such that any2-edge colouring ofGhas a monochromatic (induced) subgraphK n Note, however, that in Corollary 4.2 the monochromatic subgraph H

i

is not required to be induced

The following impressive theorem improves the results we have mentioned in this chapter and it has a difficult proof

graph Then there exists a graphGsuch that any2-edge colouring ofGhas an monochromatic induced subgraphH.

Example 4.4 As an application of Ramsey’s theorem, we shortly describe Schur’s theorem.

For this, consider the partition f1; 4; 10; 13g; f2; 3; 11; 12g; f5; 6; 7; 8; 9g of the set N

1 3

= [1; 13℄ We observe that in no partition class there are three integers such that x + y = z However, if you try to partition N

14 into three classes, then you are bound to find a class, wherex + y = zhas a solution

SCHUR(1916) solved this problem in a general setting The following gives a short proof using Ramsey’s theorem

For eachn  1, there exists an integerS(n)such that any partitionS

1

; : ; S

nofN S(n) has

a classS

i containing two integersx; ysuch thatx + y 2 S

i.

Indeed, letS(n) = R (3; 3; : : 3), where 3occurs ntimes, and letK be a complete on N

S(n) For a partitionS

1

; : ; S

nofN S(n), define an edge colouringofKby

(ij) = k; ifji jj 2 S

k :

By Theorem 4.8,K

 has a monochromatic triangle, that is, there are three vertices1  i <

j < t  S(n)such thatt j; j i; t i 2 S

k for somek But(t j) + (j i) = t i proves the claim

There are quite many interesting corollaries to Ramsey’s theorem in various parts of

math-ematics including not only graph theory, but also, e.g., geometry and algebra, see

R.L GRAHAM, B.L ROTHSCHILD ANDJ.L SPENCER, “Ramsey Theory”, Wiley, (2nd ed.) 1990

4.3 Vertex colourings

The vertices of a graphGcan also be classified using colourings These colourings tell that certain vertices have a common property (or that they are similar in some respect), if they share the same colour In this chapter, we shall concentrate on proper vertex colourings, where adjacent vertices get different colours

4 18 25 3 5 -4 1 4 9-6 1 5 5-8 4 6 9-1 15 8 0-1 49

5 14 25 4 3 -4 9 5 8-8 7 8 0-1 43 9 5-2 16 12 1-3 16 14 1 -4 42

The first unknownR (p; p)(where... , and let be an optimal -edge colouring of