(d,1)-total labelling of sparse graphs

HAL Id: hal-00401711

https://hal.archives-ouvertes.fr/hal-00401711

Submitted on 3 Jul 2009

HAL is a multi-disciplinary open access

archive for the deposit and dissemination of

sci-entific research documents, whether they are

pub-lished or not. The documents may come from

teaching and research institutions in France or

abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est

destinée au dépôt et à la diffusion de documents

scientifiques de niveau recherche, publiés ou non,

émanant des établissements d’enseignement et de

recherche français ou étrangers, des laboratoires

publics ou privés.

(d,1)-total labelling of sparse graphs

Louis Esperet, Mickael Montassier, André Raspaud

To cite this version:

(d, 1)

-total labelling of sparse graphs Louis Esperet

∗

, Mi kaël Montassier

†

and André Raspaud

‡

LaBRI UMR CNRS 5800, Université BordeauxI,

33405 Talen e Cedex

FRANCE.

28th February 2006

Abstra t

The

(d, 1)

-totalnumber

λ

T

d

(G)

ofagraph

G

isthewidthofthesmallestrangeofintegers that su es to labelthe verti esand theedges of

G

so that notwo adja entverti eshave thesame olor,notwoin identedgeshavethesame olorandthedistan ebetweenthe olor

of a vertex andthe olorof any in identedge isat least

d

. Thisnotion was introdu ed by Havetand Yu in [6℄. In thispaper, westudy the

(d, 1)

-total numberof sparse graphs and provethat forany

0 < ε <

1

2

, andanypositiveinteger

d

,there exists a onstant

Cd,ε

su h that forany

ε∆

-sparsegraph

G

withmaximumdegree

∆

,wehave

λ

T

d

(G)

≤ ∆ + C

d,ε

.

1 Introdu tion

Inthe hannelassignmentproblem, weneed to assignfrequen y bandsto transmitters. Iftwo

transmittersaretoo lose,interferen eswillo uriftheyattempttotransmiton losefrequen ies.

In orderto avoidthis situation, the hannels assignedmust be su iently far. Moreover, if two

transmitters are lose but not too lose, the hannels assigned must still be dierent. This

problem is known under the

L(p, q)

-labelling problem of a graph

G

, where a

L(p, q)

-labelling is an integer assignment

L

to the verti es of

G

su h that

∀(u, v) ∈ V (G)

2

_{, d}

G

(u, v) = 1

⇒

|L(u) − L(v)| ≥ p

and

∀(u, v) ∈ V (G)

2

_{, d}

G

(u, v) = 2

⇒ |L(u) − L(v)| ≥ q

. In1992, Griggs and Yeh introdu ed this labelling with

p = 2

and

q = 1

in [4 ℄. Sin e, this notion has been widely studied and gives many hallenging problems. In parti ular, in 1995, Whittlesey, Georges and

Mauro [3 ℄ studied the

L(2, 1)

-labelling of the in iden e graph obtained from

G

. The in iden e graph of

G

is the graph obtained from

G

by repla ing ea h edge by a path of length 2. The

L(2, 1)

-labelling of the in iden e graph of

G

is equivalent to an assignment of integers to ea h elementof

V (G)

∪ E(G)

su h that:

1. theedge- oloring isproper, i.e. notwo in ident edgesre eive thesame integer;

2. thevertex- oloring isproper, i.e. no twoadja ent verti es re eivethesame integer ;

3. the dieren e between the integer assigned to a vertex and those assigned to its in ident

Thislabellingis alleda

(2, 1)

-totallabelling. It was introdu ed byHavetand Yuin2002 [6 , 5 ℄ and generalized to the

(d, 1)

-totallabellingof a graph

G

.

More formally, a

(d, 1)

-total labelling of a graph

G = (V, E)

is a fun tion

c : V

∪ E → N

verifying:

(i)

_{∀(u, v) ∈ V}

2

: uv

_{∈ E ⇒ c(u) 6= c(v)}

(ii)

_{∀(u, v, w) ∈ V}

3

_{: uv}

_{∈ E, uw ∈ E ⇒ c(uv) 6= c(uw)}

(iii)

∀(u, v) ∈ V

2

_{: uv}

_{∈ E ⇒ |c(u) − c(uv)| ≥ d}

Thespan ofa

(d, 1)

-totallabellingisthemaximumdieren ebetween twoassignedintegers. The

(d, 1)

-total number of a graph

G

,denoted by

λ

T

d

(G)

, isthe minimumspan of a

(d, 1)

-total labellingof

G

. Figure1givesanexampleofa

(2, 1)

-totallabellingwith6 olors(weuseintegers belongingto anintervalbeginning byzero).

0

0

0

5

3

3

5

1

1

1

4

3

5

3

5

1

1

4

4

5

2

0

5

3

2

Figure1:

(2, 1)

-totallabellingof thePetersen's graph.

Noti ethatthe

(1, 1)

-totallabellingis thetraditionaltotal oloring.

We re allsome bounds (withoutproof) and a onje ture for the

(d, 1)

-total number: Theorem 1 ([6℄) Let

G

be a graph with maximum degree

∆

, then:

(i) λ

T

d

(G)

≥ ∆ + d − 1

.

(ii)

If

G

is

∆

-regular,

λ

T

d

(G)

≥ ∆ + d

.

(iii)

If

d

≥ ∆

,

λ

T

d

(G)

≥ ∆ + d

. Let

χ(G)

(resp.

χ

′

_(G)

)bethe hromati number(resp. index)of

G

. Observethatifwe olorthe verti es with olorsbelongingto aninterval

I

v

ontaining

χ(G)

olorsandtheedgeswith olors belonging to an interval

I

e

ontaining

χ

′

_(G)

olors,

I

v

and

I

e

being separated by an interval of size

d

− 1

, we obtain a

(d, 1)

-total labelling of the graph. Theorem 2 is dedu ed from this observation :

Theorem 2 ([6℄) Let

G

be a graph,then

(i) λ

T

d

(G)

≤ χ(G) + χ

′

(G) + d

− 2

(ii) λ

T

_d

(G)

≤ 2∆ + d − 1

Theorem 3 ([10℄) Let

G

be a onne ted graph with maximum degree

∆

,

d

≥ 2

, then

λ

T

d

(G)

≤

(i) ∆

_{≥ 2d + 1}

and

M ad(G) <

5

2

(ii) ∆

_{≥ 2d + 2}

and

M ad(G) < 3

(iii) ∆

≥ 2d + 3

and

M ad(G) <

10

3

where

M ad(G)

isthemaximum averagedegree of

G

, i.e.

M ad(G) = max

{2|E(H)|/|V (H)|, H j

G

}

.

Conje ture 1 ([6℄) Let

G

be a graph with maximum degree

∆

, then

λ

T

d

(G)

≤ min{∆ + 2d −

1, 2∆ + d

_{− 1}}

.

Finally,thebestknownupperbound for generalgraphs isdueto Esperetand Havet[2 ℄ who

proved :

Theorem 4 Let

G

be a graph with maximum degree

∆

, then

λ

T

d

(G)

≤ ∆ + O(log ∆)

.

In [7 ℄, Molloy and Reed proved that the total hromati number of any graph with maximum

degree

∆

is at most

∆

plus an absolute onstant. Moreover, in [9 ℄, they gave a simpler proof of this result for sparse graphs. In this paper, we generalize their approa h to the

(d, 1)

-total numberofsparse graphs.

A vertex

v

is alled

α

-sparse i

|E(N(v))| ≤

∆

2

− α∆

. An

α

-sparse graph is a graph in whi h all theverti esare

α

-sparse.

Ourmain resultisthefollowing :

Theorem 5 For any

0 < ε <

1

2

, and any positive integer

d

, there exists a onstant

C

d,ε

su h that for any

ε∆

-sparse graph

G

with maximum degree

∆

, we have

λ

T

d

(G)

≤ ∆ + C

d,ε

.

The proof of Theorem 5 is based on a probabilisti approa h due to Molloyand Reed. It uses

intensively on entration inequalitiesas well as LovászLo al Lemma. Moreover,we onje ture:

Conje ture 2 For any positive integer

d

, there existsa onstant

C

d

,su h that for any graph

G

with maximum degree

∆

,we have

λ

T

d

(G)

≤ ∆ + C

d

.

In Se tion 2,we present thepro edure used to prove Theorem 5. In Se tion 3, we analyse this

pro edure. In the following, we will need some probabilisti tools (see Appendix

A

and [9 ℄ for more details).

2 Proof of Theorem 5

Sin e

λ

T

d

(G)

≤ 2∆ + d − 1

,if we prove thatfor some

∆

0

(d, ε)

and some

C

d,ε

,any

ε∆

-sparse graph

G

ofmaximumdegree

∆

≥ ∆

0

veries

λ

T

d

(G)

≤ ∆ + C

d,ε

,thenTheorem5 willbe proved.

Let

φ

be afull or partial oloringof

G

. Anyedge

e = uv

su hthat

|φ(u) − φ(e)| < d

or/and

To proveTheorem 5,we apply the following steps:

Step 1. First,wewill olor the edgesbyVizing's Theorem usingthe olors

{1, . . . , ∆}

.

Step 2. ThenwewillusetheNaiveColoringPro edureto olortheverti eswith olors

{1, . . . , ∆+

2d

_{− 1}}

. Thispro edure reatesreje t edges. However,we an prove thatafter the pro e-dure, themaximumdegreeofthereje tgraph

R

isa onstant

D

d,ε

whi h doesnotdepend on

∆

.

Step 3. Finally,we erase the olor ofthe verti es of

R

and re olorthese verti es greedily withthe olors

{∆ + 3d − 2, . . . , ∆ + 3d − 1 + D

d,ε

}

. Taking

C

d,ε

= D

d,ε

+ 3d

− 2

,thisproves that

λ

T

d

(G)

≤ ∆ + C

d,ε

.

We now present the Naive Coloring Pro edure.

2.2 The Naive Coloring Pro edure

For ea h vertex

v

, we maintain two lists of olors:

L

v

and

F

v

.

L

v

is the setof olorswhi h do not appear inthe neighborhood of

v

. Initially,

L

v

=

{1, . . . , ∆ + 2d − 1}

. After iteration

I

(spe ied later),

F

v

will bea setof forbidden olors. Until iteration

I

,

F

v

=

∅

.

During the Naive Coloring pro edure, we will perform

i

∗

(spe ied later) iterations of the

following pro edure:

Step 1. Assignto ea h un olored vertex

v

a olor hoosenuniformlyat random in

L

v

. Step 2. Un olor anyvertexwhi h re eives thesame olor as a neighbor inthisiteration.

Step 3. Iteration

i

≤ I

. Let

v

be a vertex having more than

T

(spe ied later) neighbors

u

whi h areassigned a olor

c(u)

su h that

|c(uv) − c(u)| < d

inthis iteration. Forany

v

,we un olor allsu h neighbors. Iteration

i > I

.

(a) Un olor anyvertex

v

whi hre eivesa olorfrom

F

v

inthis iteration.

(b) Let

v

beavertexhavingmorethan oneneighbor

u

whi hisassigneda olorsu h that

|c(uv) − c(u)| < d

inthis iteration. For any

v

,we un olorall su h neighbor. ( ) Let

v

be a vertex having at least one neighbor

u

su h that

|c(uv) − c(u)| < d

in this iteration. For any

v

,wepla e

{c(vw) − d + 1, . . . , c(vw), . . . , c(vw) + d − 1}

in

F

w

for every

w

∈ N(v)

.

Step 4. Foranyvertex

v

whi h retainedits olor

c

,we remove

c

from

L

u

for any

u

∈ N(v)

.

After

i

∗

iterationsof this pro edure, wehave have apartial oloring of

G

. We then omplete this oloring in order to obtain a reje t graph

R

witha bounded maximum degree whi h does not depend on

∆

(Se tion 4.3).

3 Analysis of the pro edure

3.1 The rst iteration

Let

ζ =

ε

2e

3

. In this subse tion,we prove that:

Claim 1 Therstiterationprodu esapartial oloringwithboundedreje tdegreeforwhi hevery

vertex has at least

ζ

We re allthat

C = ∆ + 2d − 1

is theinitial size ofea h olor list

L

v

. Let

A

v

be thenumber of olors

c

su h that at least two neighbors of

v

re eive the olor

c

and all su h verti es retain their olor during Step2. Let

B

v

be the numberof neighbors of

v

whi h areun olored at Step 3. Noti ethat verti es are un olored at Step3 regardless of what happened at Step 2. Let

X

v

be theevent that 

A

v

< ζ∆

. Let

Y

v

be theevent that 

B

v

≥

ζ

2

∆

. If notype

X

event o urs, everyvertexhas atleast

ζ∆

repeated olorsinitsneighborhood attheendof Step2. Ifnotype

Y

event o urs, less than

ζ

2

∆

verti es areun olored inea hneighborhood. Asa onsequen e,if weshow thatwithpositive probability,no type

X

or

Y

event o urs,Claim1 willbeproved.

Claim 2

Pr(X

v

) < e

−α log

2

∆

,for a parti ular onstant

α > 0

.

Proof. We rst bound the expe ted value of

A

v

. Let

A

′

v

be the number of olors

c

su h that exa tly two neighbors of

v

re eive the olor

c

and arenot un olored during Step2. Noti e that

A

v

≥ A

′

v

, and thus

E(A

v

)

≥ E(A

′

v

)

. Let

u

and

w

be two non adja ent neighbors of

v

. The probability that

u

and

w

are olored with

c

,while noother neighbor of

v

is olored with

c

,and while no neighbor of

u

or

w

is olored with

α

is exa tly

1

C

2

1

₋

_C

1

3∆−3

>

1

_C

2

1

₋

1

_C

3∆

.

Sin e

G

is

ε∆

-sparse,

|E(N(v))| ≤

∆

2

− ε∆

2

,whi himpliesthatthere areatleast

ε∆

2

pairsof

non adja ent verti es amongthe neighbors of

v

. Thereare

C

hoi es for the olor

c

,thus

E(A

′

_{v) >}

_Cε∆

2

 1

C

2



1

−

1

C

3∆

=

ε∆

2

C



1

−

1

C

3∆

For

∆ > 2

, we have

ln(1

−

1

C

)

≥ −

C

1

−

C

1

2

,and thus

1

−

1

C

3∆

≥ e

−3

e

−

3

C

. For

∆

large enough,

∆/

C >

√

3/2

and

e

−

3

C

>

√

3/2

,so:

E(A

′

v

) >

3ε∆

4e

3

=

3

2

ζ∆

Sin e

E(A

v

)

≥ E(A

′

v

)

, we also have

E(A

v

) >

3

2

ζ∆

. Let

AT

v

be thenumber of olors assigned to at least two neighbors of

v

, and let Del

v

be the number of olors assigned to at least two neighbors of

v

and not retained byat least one of them. Note that

A

v

= AT

v

−

Del

v

, and by linearity of expe tation,

E(A

v

) = E(AT

v

)

− E(

Del

v

)

. The random variable

AT

v

only depends on the

∆

olorsassigned to the neighbors of

v

. Moreover, hangingone of these olors an only ae t

AT

v

byat most 1. Using the SimpleCon entrationbound,we obtain:

Pr (|AT

v

− E(AT

v)

| > t) < 2e

−

t2

2∆

.

(1)

The random variable Del

v

only depends on the nearly

∆

2

olors assigned to the verti es at

distan e at most 2 from

v

. As previously, hanging one of these olors an only ae t Del

v

by at most 1. Furthermore, if Del

v

≥ s

, we an nd at most

3s

verti es, whi h olors ertify that Del

v

≥ s

(forea h olor

α

ountedbyDel

v

≥ s

,wetaketwoneighbors

x

and

y

of

v

olored with

α

and a neighbor

z

of

x

or

y

also olored with

α

). Applying Talagrand's Inequality with

c = 1

and

r = 3

,we obtainfor all

t

≥

√

∆ log ∆

Pr (|

Del

v

− E(

Del

v)

| > t) < 4e

−

(

t−60

√

3E(Delv )

)

2

24E(Delv )

_{< 4e}

−

25∆

t2

,

(2)

sin e

E(

Del

v

)

≤ ∆

. Re all that

E(A

v

) = E(AT

v

)

− E(

Del

v

)

. Let

t =

1

2

log ∆

pE(A

v

)

. If

|A

v

− E(A

v

)

| > log ∆pE(A

v

)

wehaveeither

|AT

v

− E(AT

v

)

| > t

or

|

Del

v

− E(

Del

v

)

| > t

. Using (1)and (2), the probabilitythatthis happensisat most

So,for

∆

large enough,

Pr



|A

v

− E(A

v

)

| > log ∆pE(A

v

)



< e

−

100

ζ

log

2

∆

.

Pr



|A

v

− E(A

v)

| > log ∆

p

E(Av)



≥ Pr



Av

< E(Av)

− log ∆

p

E(Av)



≥ Pr



Av

<

3

2

ζ∆

− log ∆

√

∆



≥ Pr (A

v

< ζ∆)

Sin e

Pr(X

v

) = Pr(A

v

< ζ∆)

,we proved that

Pr(X

v

) < e

−

₁₀₀

ζ

log

2

∆

.

2

Claim 3

Pr(Y

v

) < e

−β∆

, for a parti ular onstant

β > 0

.

Proof. Let

u

bea neighborof

v

. Thevertex

u

will be un olored inStep3if forsome neighbor

w

of

u

,

u

and

T

other neighbors

x

1

, . . . , x

T

of

w

are ea h assigned a olor

c(x

i

)

su h that

|c(u) − c(wu)| < d

and

|c(x

i

)

− c(wx

i

)

| < d

forall

1

≤ i ≤ T

. Theprobabilitythat thishappens is at most

∆

∆ − 1

T

  2d − 1

C



T+1

<

(2d

− 1)

T

T !

For

T

large enough,

(2d

− 1)

T

_{/T ! < ζ/4}

,and thus

E(B

v

) <

ζ∆

4

. The random variable

B

v

only dependsonthenearly

∆

3

olorsassignedto theverti es atdistan eatmost 3from

v

. Changing one ofthese olors an ae t

B

v

byat most

T + 1

. Moreover,if

B

v

≥ s

thereisa setofat most

(T + 1)s

verti es whi h olors ertify that

B

v

≥ s

(for ea h un olored neighbor

u

of

v

, take

u

and

T

other neighbors

x

1

, . . . , x

T

of some neighbor

w

of

u

, su h that

|c(u) − c(wu)| < d

and

|c(x

i

)

− c(wx

i

)

| < d

for all

1

≤ i ≤ T

). Applying Talagrand's Inequality to

B

v

with

c = T + 1

and

r = T + 1

,weobtain for all

t

≥

√

∆ log ∆

Pr (

|B

v

− E(B

v)

| > t) < 4e

−

(

t−60(T +1)

√

(T +1)E(Bv )

)

2

8(T +1)3E(Bv )

_{< 4e}

−

t2

9(T +1)3∆

_.

Taking

t =

ζ∆

8

,weobtain

Pr



|B

v

− E(B

v

)

| >

ζ∆

₈



< 4e

−

ζ2∆

576(T +1)3

_{< e}

−

ζ2∆

577(T +1)3

. Now, sin e

Pr



|B

v

− E(B

v

)

| >

ζ∆

8



≥ Pr



Bv

> E(Bv) +

ζ∆

8



≥ Pr



Bv

>

3

8

ζ∆



≥ Pr



Bv

≥

ζ∆

₂



we have

Pr(Y

v

) < e

−

_{577(T +1)3}

ζ2

∆

.

2

We now use Lovász Lo al Lemma to prove Claim 1. Ea h event

X

v

only depends on the olors assigned to the verti es at distan e at most 2 from

v

,and ea h event

Y

v

depends on the olors assigned to the verti es at distan e at most 3 from

v

. Hen e, ea h event is mutually independent of all but at most

2∆

6

other events. For

∆

su iently large,

Pr(X

v

) <

1

8∆

6

and

Pr(Y

v

) <

_8∆

1

6

. Using Lovász Lo al Lemma, this proves that with positive probability no type

X

or

Y

event happens. Thus with positive probability, the rst iteration produ es a partial oloring withbounded reje tdegree,su h thatea hvertexhas at least

ζ∆

Let

d

i

=



1

₋

1

₄

e

−

2

ζ



i

∆

and

f

i

=

4(2d−1)

ζ

P

i−1

j=I+1

D

j

. Let

i

∗

be thesmallest integer

i

su h that

d

i

≤

√

∆

. Observe thatfor any

i

≤ i

∗

,wehave

d

i

≥ (1 −

1

4

e

−

2

_ζ

)

√

∆

. Claim 4 At the end of ea h iteration

1

≤ i ≤ i

∗

, with positive probability every vertex has at

most

d

i

un olored neighbors, andea h list

F

v

has size atmost

f

i

.

Proof. We prove Claim 4 by indu tion on

i

. At the end of the rst iteration, every vertex has at least

ζ∆

2

repeated olors inits neighborhood. Sothe numberof un olored verti es inthe neighborhood of any vertex is at most

(1

− ζ)∆

, whi h is less than

d

1

=



1

₋

1

₄

e

−

ζ

2



∆

. Mor-ever,foranyvertex

v

,thelist

F

v

isstillemptyattheendoftherstiteration,thus

|L

v

| ≤ 0 = f

1

.

Suppose

i > 1

. Byindu tion, thereareatmost

d

i−1

un olored verti esinea hneighborhood atthe beginning ofiteration

i

,andea h

F

v

has sizeatmost

f

i−1

. Wedene therandomvariable

D

i

v

as the numberof un olored neighbors of

v

after iteration

i

, and the random variable

F

i

v

as the size of the list

F

v

after iteration

i

. To omplete the indu tion, we show that with positive probability,

D

i

v

≤ d

i

and

F

i

v

≤ f

i

for anyvertex

v

. Sin eeveryvertex

v

hasat least

ζ∆

2

repeated olorsinitsneighborhood,everylist

L

v

has sizeat least

ζ∆

2

. Thus, theprobabilitythatanewly olored vertex is not un olored during Step 2 is at least



1

₋

_ζ∆

2



∆

. So the probability thata newly olored vertexis un olored duringStep2 isat most:

1

₋



1

₋

2

ζ∆

∆

≤ 1 −

3

₄

e

−

2

ζ

For

i

≤ I

,theprobabilitythatthenewly oloredvertex

v

isun oloredduringStep3isat most:

∆

d

i−1

T

  2d

_{− 1}

ζ∆/2

T

+1

≤

 2(2d

_ζ∆

− 1)

T+1

1

T !

≤

1

4

e

−

2

ζ

Observe thatfor

I

su iently largeinterms of

ζ

and

d

,we have

fi

=

4(2d

− 1)∆

ζ

i−1

X

j=I+1



1

−

1

₄

e

−

2

ζ



j

≤

4(2d

− 1)∆

_ζ

× 4e

2

ζ



1

−

1

₄

e

−

2

ζ



I+1

<

ζ∆

16

e

−

2

ζ

.

Thus, for

i > I

,the probabilitythat thevertex

v

isun olored during Step3

(a)

isat most:

|F

v

|

|L

v

|

≤

2

ζ∆

f

i−1

<

1

8

e

−

2

ζ

Andthe probability that

v

isun olored during Step3

(b)

is atmost:

∆d

i−1

 2(2d − 1)

_ζ∆

2

≤



1

−

1

₄

e

−

2

ζ

I

 2(2d − 1)

ζ

2

≤

1

₈

e

−

ζ

2

Combiningtheseresults,theprobabilitythatanewly oloredvertexisun olored during Step2

Let

X

i

v

be the event that

D

i

v

>



1

₋

1

₄

e

−

2

ζ



d

i−1

. We dene the random variable

N F

i

v

as the numberof olors added to

F

v

during iteration

i

. Let

Y

i

v

be theevent that

N F

i

v

>

4(2d−1)

ζ

d

i−1

. Using Lovász Lo al Lemma, we prove that with positive probability none of the type

X

or

Y

events o urs. Claim 5

Pr(X

i

v

) < e

−δ log

2

_d

i−1

, for a parti ular onstant

δ > 0

.

Proof. Let

v

beavertexof

G

. Let

A

bethenumberofneighborsof

v

thatareun olored during Step 2. For

i

≤ I

we dene

B

as thenumberof neighbors of

v

thatare un olored during Step 3. For

i > I

we dene

C

(resp.

D

) as the number of neighbors of

v

that are un olored during Step

3.(a)

(resp.

3.(b)

). Using the Simple Con entration Bound on

A

, Talagrand's Inequality on

B

and

D

,andCherno Boundon

C

, ombinedwith

E(D

i

v

)

≤ (1 −

1

₂

e

−

2

ζ

_)d

i−1

,we prove the following inequalities:

Pr



|A − E(A)| >

1

₂

log d

i−1

p

E(A + B)



< 2e

−

e

−

2

ζ

64

log

2

_d

i−1

(3)

Pr



|B − E(B)| >

1

₂

log d

i−1

p

E(A + B)



< 4e

−

e

−

_ζ

2

64(T +1)3

log

2

_d

i−1

(4)

Pr



|A − E(A)| >

1

₃

log d

i−1

p

E(A + C + D)



< 2e

−

e

−

2

ζ

144

log

2

_d

i−1

(5)

Pr



|C − E(C)| >

1

₃

log d

i−1

p

E(A + C + D)



< 2e

−

144

1

log

2

_d

i−1

(6)

Pr



|D − E(D)| >

1

₃

log d

i−1

p

E(A + C + D)



< 2e

−

e

−

2

_ζ

1152

log

2

_d

i−1

(7)

The proof of these results is very lose fromthe proofs ofClaims 2 and 3. Combining (3), (4),

(5), (6)and (7), we obtainfor

T

and

∆

large enough :

Pr(X

v

i

) < e

−

e

−

_ζ

2

65(T +1)3

log

2

_d

i−1

2

Claim 6

Pr(Y

i

v

) < e

−γd

i−1

, for a parti ular onstant

γ > 0

.

Proof. Theprobabilitythataneighbor

u

of

v

isassigneda olor

c(u)

su hthat

|c(u)−c(uv)| < d

is

2d−1

|L

u

|

≤

2(2d−1)

ζ∆

. Thus

E(N F

v

)

≤

2(2d−1)

ζ∆

d

i−1

. ApplyingTalagrand'sInequalityto therandom variable

N F

v

with

c = (2d

− 1)

2

and

r = 1

,we obtain :

Pr (

|NF

v

− E(NF

v)

| > t) < 4e

−

_{16(2d−1)5di−1}

ζt2

for any

t > log d

i−1

pd

i−1

. Taking

t =

2d−1

ζ

d

i−1

,we obtain :

Pr



N Fv

>

4(2d

− 1)

ζ

d

i−1



≤ Pr



|NF

v

− E(NF

v)

| >

2d

_{− 1}

ζ

d

i−1



< 4e

−

2ζ(2d−1)3

di−1

2

Thevariable

X

i

v

onlydependsonthe olorsassignedtotheverti esatdistan eatmost3from

v

duringiteration

i

,whilethevariable

Y

i

v

dependsonthe olorsassignedtotheverti esatdistan e at most 2 from

v

during iteration

i

. Thus, ea h type

X

or

Y

event is mutually independant from all but at most

2d

6

i−1

other events. Using Claims 5 and 6, we have

Pr(X

i

v

) <

_8d

1

6

i−1

and

Pr(Y

v

i

) <

_8d

1

6

i−1

for

∆

large enough (re all that a ording to our hoi e of

i

∗

we always have

d

i

≥ (1 −

1

₄

e

−

2

ζ

₎

√

_∆

At this point, wehave a partial oloring su hthat:

•

ea h vertex

v

has at most

√

∆

un olored neighbors;

•

thereje t degree ofea h vertexisat most

IT + 1

;

•

ea h vertexhas a listofat least

ζ∆

2

available olors.

Itwillbemore onvenientto uselistsofequalsizes. Sowearbitrarily delete olorsfromea h

list, so that for every un olored vertex

v

, we have

|L

v

| =

ζ∆

2

. For ea h un olored vertex, we hooseasubsetof olorsfrom

L

v

whi hwill be andidates for

v

andwe provethatwithpositive probability, there exists a andidate for ea h un olored vertex, su h that we an omplete our

partial oloringof

G

.

A andidate

a

for

v

is saidto begood if:

Condition 1 for every neighbor

u

of

v

,

a

isnot andidate for

u

;

Condition 2 for every neighbor

u

of

v

,and every neighbor

w

of

u

,there is no andidate

b

of

w

su hthat

|c(uv) − a| < d

and

|c(uw) − b| < d

.

Ifwe nd a good andidate for every un olored vertex, Condition 1 ensures thatthe vertex

oloringobtainedisproper,andCondition2ensuresthatnoreje tdegreein reasesbymorethan

one.

Claim 7 There exists a set of andidates

S

v

for ea h un olored vertex

v

, su h that ea h set ontains atleast one good andidate.

Proof. For ea h un olored vertex

v

, we hoose a random permutation of

L

v

, and take the rst twenty olors of the list as set of andidates for

v

. Let

C

v

be the event that none of the andidates for

v

isagood andidate. Ea hevent

C

v

dependsfromat most

∆

4

otherevents. We

nowshow that

Pr(C

v

) <

1

4∆

4

. LovaszLo alLemmawill omplete theproof.

Let

v

be an un olored vertexof

G

. We dene:

Bad

1

=

{c ∈ L

v

: c

is andidate for some neighbor of

v

}

Bad

2

=

{c ∈ L

v

:

hoosing

c

for

v

violates Condition2

}

Bad = Bad

1

∪ Bad

2

Let

D

be the event that

|Bad| ≤ 60(2d − 1)

2

√

_∆

. A andidate for

v

isgood if and only if it doesnot belong to

Bad

. Observethat :

Pr(Cv

|D) ≤



|Bad|

|L

v

|

20

≤

60(2d

− 1)

2

√

_∆

ζ∆

2

!20

≤

120

20

_(2d

− 1)

40

ζ

20

_∆

10

Sofor

∆

su iently large,

Pr(C

v

|D) <

1

8∆

4

.

Ea h vertex has at most

√

∆

un olored neighbors, thus

|Bad

1

| ≤ 20

√

∆

_{≤ 20(2d − 1)}

2

√

∆

. We now show that with very high probability, the size of

Bad

2

is at most

40(2d

− 1)

2

√

_∆

. A

olor

c

belongs to

Bad

2

if for someneighbor

u

of

v

su h that

|c(uv) − c| < d

,there isaneighbor

w

of

u

and a andidate

a

for

w

su h that

|c(uw) − a| < d

. Thus weobtain:

Pr(c

∈ Bad2)

≤ (2d − 1) × 20

√

∆

×

2d

ζ

− 1

∆

2

E(|Bad2|) ≤

ζ∆

₂

×

40(2d

− 1)

2

ζ

√

∆

≤ 20(2d − 1)

2

√

_∆

The random variable

|Bad

2

|

only depends on at most

∆

2

permutations of olor lists of

un olored verti es at distan e at most 2 from

v

. Moreover, ex hanging two members of one of the permutations an ae t

|Bad

2

|

byat most

2d

− 1

. If

|Bad

2

| ≥ s

, we an ertify this by giving, forea h olor

α

∈ Bad

2

,aneighbor

u

of

v

su hthat

|c(uv) − α| < d

,aswellasaneighbor

w

of

u

having a andidate

a

su h that

|c(uw) − a| < d

. Re all that

a

is a andidate for

w

if it belongsto therst twentypositions ofthepermutationof

L

w

. So weonlyneed togive

s

hoi es of andidates to ertifythat

|Bad

2

| ≥ s

. We apply M Diarmid'sInequalityto

X =

|Bad

2

|

with

n = 0

,

m = ∆

2

,

c = 2d

− 1

,

r = 1

,and

t = 10(2d

− 1)

2

√

_∆

:

Pr



|X − E(X)| > 10(2d − 1)

2

√

_{∆ + 60(2d}

− 1)

p

E(X)



< 4e

−

100(2d−1)4∆

8(2d−1)2E(X)

Sin e

E(X)

≤ 20(2d − 1)

2

√

_∆

,this impliesfor

∆

su iently large:

Pr



|Bad2| > 40(2d − 1)

2

√

_∆



_{< 4e}

−

5

8

√

∆

So for

∆

large enough,

Pr D

<

_8∆

1

4

. We an express the probability of

C

v

as

Pr(C

v

) =

Pr(C

v

|D)Pr(D) + Pr(C

v

|D)Pr(D)

. Andso,

Pr(Cv)

_{≤ Pr(C}

v

|D) + Pr(D) <

1

4∆

4

2

We obtaina oloring of

G

with maximumreje tdegree at most

IT + 2

. So thereje t graph

R

obtained has maximumdegree atmost

IT + 2d + 1

. Weun olor theverti esof

R

and re olor them greedilywiththe olors

{∆ + 3d − 2, . . . , ∆ + IT + 5d}

. Thisnal oloring isa

(d, 1)

-total labelling of

G

. Sin e

I

and

T

areindependant of

∆

,we provedthat

λ

T

d

(G)

≤ ∆ + C

d,ε

.

Remark 1 By looking arefullyat ea h inequality during thepro edure, we anrepla e

∆ + C

d,ε

by

∆ + C

ε

d log d

, where

C

ε

isa onstant that does not depend on

d

.

4 Further work

Theorem 5 an be transformed into a randomized algorithm, using a powerful te hnique

intro-du ed by Be k [1℄ and extended to a wide range of appli ations of the symmetri form of the

Lo alLemmabyMolloyandReed [8℄.

Referen es

[1℄ Be kJ.AnAlgorithmi Approa htotheLovászLo alLemmaI,RandomStru tures&Algorithms,

2:343365,1991.

[2℄ EsperetL.Coloration

(d, 1)

-totaleetméthodeprobabiliste, Master'sthesis,ENSLyonandINRIA SophiaAntipolis,2003.

[3℄ Georges J.P., Mauro D.W., and Whittlesey M.A. On the

λ

-number of Q

n

and related graphs, SIAM, J.Dis reteMath,8:449506,1995.

[4℄ Griggs J.R. and YehR.K. Labellinggraphs with a onditionat distan e two,SIAM, J. Dis rete

[5℄ Havet F.

(d, 1)

-total labelling of graphs, Workshop Graphs and Algorithms, Dijon (FRANCE), 2003.

[6℄ HavetF.andYu M.-L.

(d, 1)

-totallabellingofgraphs, Te hni alReport4650,INRIA,2002. [7℄ MolloyM.andReedB. Aboundonthetotal hromati number, Combinatori a,2:241280,1998.

[8℄ MolloyM.andReedB.FurtherAlgorithmi Aspe tsoftheLo alLemma,Pro eedingsofthe30th

ACMSymposiumonTheoryofComputing,1998.

[9℄ MolloyM. andReedB. Graph oloring andthe probabilisti method, Vol.23 ofAlgorithms and

ombinatori s,Springer-Verlag,2002.

[10℄ Montassier M. and Raspaud A.

(d, 1)

-total labelling of graphs with a given maximum average degree, JournalofGraphTheory,2005,toappear.

A Probalisti tools

Simple Con entration Bound. Let

X

be a random variable determined by

n

independent trials

T1, . . . , Tn

andsatisfying:

1. Changing the out omeof anyone trial an ae t

X

byatmost

c

. Then,

Pr(

|X − E(X)| > t) ≤ 2e

−

2c2n

t2

Talagrand's Inequality. Let

X

be anon-negativerandom variable, not identi ally 0, whi h is deter-minedby

n

independenttrials

T1, . . . , Tn

,

andsatisfyingthe following forsome

c, r > 0

:

1. Changing the out omeof anyone trial an ae t

X

byatmost

c

.

2. Forany

s

,if

X

≥ s

thenthereisaset ofatmost

rs

trials whoseout omes ertifythat

X

≥ s

. Thenforany

0

≤ t ≤ E(X)

,

Pr



|X − E(X)| > t + 60c

prE(X)



≤ 4e

−

8c2rE(X)

t2

M Diarmid's Inequality. Let

X

beanon-negativerandomvariable,not identi ally 0,whi h is deter-mined by

n

indenpendent trials

T1, . . . , Tn

and

m

independent permutations

Π1, . . . , Πm

and satisfying the followingforsome

c, r > 0

:

1. Changing the out omeof anytrial an ae t

X

byatmost

c

.

2. Inter hangingtwoelements inany onepermutation anae t

X

by atmost

c

.

3. Forany

s

,if

X

≥ s

thenthereisaset ofatmost

rs

hoi eswhose out omes ertifythat

X

≥ s

. Thenforany

0

≤ t ≤ E(X)

,

Pr



|X − E(X)| > t + 60c

prE(X)



≤ 4e

−

8c2rE(X)

t2

LovászLo alLemma. Consider aset

E

of(typi ally bad) eventssu hthat forea h

A

∈ E

1.

Pr(A)

≤ p < 1

,and