The L(2,1)-labeling problem on graphs

THE

L(2,

1)-LABELING

PROBLEM

ON GRAPHS*

GERARD J.

CHANGe

AND DAVID

KUO

Abstract. An L(2,1)-labelingofagraph Gis a function _f from the vertex set V(G) tothe

set of all nonnegative integers such that

If(x)-

f(Y)l

->

2 ifd(x,y) 1 and If(x) f(Y)l

->

i if

d(x,y) 2. The_5(2,1)-labeling number/k(G) ofG isthe smallest number k such that G has an

5(2,1)-labelingwith max{f(v):vE V(G)} k. Inthis paper,wegive exact formulas ofA(G2H)

and A(G

+

H). Wealso prove that A(G)

_

A2

₊

_{A for anygraph G ofmaximumdegree A. For}

odd-sun-free(OSF)-chordalgraphs,the upper boundcanbereduced toA(G)

_

2A

₊

1. Forsun-free

(SF)-chordalgraphs, the upper boundcanbe reduced toA(G)

_

A

₊

_2X(G 2. Finally,wepresent

apolynomialtime algorithmto determineA(T)foratreeT.

Key words. _L(2,1)-labeling,T-coloring, union, join, chordalgraph, perfect graph, tree,

bipar-titematching, algorithm

AMS subjectclassifications. 05C15, 05C78

1. Introduction. The channel assignmentproblemis to assign achannel

(non-negative

integer)

toeachradio transmittersothatinterferingtransmitters areassigned channels whose separation is not in aset of disallowed separations. Hale

[11]

formu-lated this problem into thenotion ofthe T-coloring ofa graph, and the T-coloring problem has been extensively studied overthe past decade

(see

[4,

5,

7,

13,

14, 16,

17,

19]).

Roberts

[15]

proposed a variation ofthe channel assignment problem in which "close" transmitters must receive different channels and

"very

close" transmitters

must receivechannelsthatare at leasttwochannels apart.

To

formulate the problem in graphs, the transmitters are represented by the vertices of a graph; two vertices are

"very

close" ifthey are adjacent in the graph and "close" ifthey are of distance twointhegraph.

More

precisely, an

L(2,

1)-labeling

ofagraph

G

is afunction

_f

from the vertex set

V(G)

to the set ofall nonnegative integers such that

If(x)- f(Y)l

>-

2 if

d(x,

y)

1 and

_If(x)

_f(Y)l

->

1 if

d(x,

y)

2.

A

k-L(2,

1)-labeling

is an

_L(2,

1)-labelingsuchthatnolabelisgreater thank. The

L(2,

1)-labeling

numberof

G,

denoted by

A(G),

is the smallest numberk such that

G

has a

k-L(2,

1)-labeling.

Griggs and Yeh

[10]

and Yeh

[21]

determined the exact values of

A(P),

A(C),

and

A(W),

where

_P

is a path of n vertices,

Cn

is a cycle of n vertices, and

_Wn

is an n-wheel obtained from

_Cn

by adding a new vertex adjacent to all vertices in

C.

For

the n-cube

Q,

Jonas

[12]

showed that n

+

3

_<

A(Q).

Griggsand Yeh

[10]

showed that

i(Qn)

_<

2n

₊

1 for n

_>

5. They also determined

A(Q)

for n

_<

5 and conjecturedthat thelowerboundn

+

3isthe actual valueof

A(Q)

forn

_>

3. Using

acodingtheory

method,

Whittlesey,

Georges,

and

Mauro

[20]

proved that

/(Qn)

_

2k

₊

2k-q+1

2,

where n

_<

2k-q and 1

_<

q

_<

k

₊

1.

In

particular,

A(Q2k-k-1)

_<

2}

1.

As

aconsequence,

A(Qn)

_<

2nfor n

_>

3.

Receivedbythe editors March 10, 1993; acceptedfor publication (in revisedform) August 1,

1995.

DepartmentofApplied Mathematics,National ChiaoTungUniversity, Hsinchu30050,Taiwan,

Republicof China(gjchang@math.nctu.edu.tw). Thisresearchwassupportedinpartbythe National

Science Councilundergrant NSC82-0208-M009-050. The researchof the first authorwassupported

inpartbyDIMACS.

309

310 GERARD J. CHANG AND DAVID KUO

For

atree

T

withmaximumdegree

A

>

1, Griggs andYeh

[10]

showedthat

A(T)

iseither

A

+

1or

A

+

2. They provedthat the

L(2,

1)-labeling

problemisNP-complete for general graphsandconjecturedthat theproblem is also NP-complete for trees.

For

a general graph

G

ofmaximum degree

A,

Griggs and Yeh

[10]

proved that

A(G) <

A2+

2A. The upperboundwasimprovedto be

A(G)

<

A2+

2A-3when

G

is

3-connected and

A(G) <

A

2 _when

_G

_{is ofdiameter two. Griggs andYeh conjectured}

that

A(G) <

A

2 _ingeneral.

_To

_{studythis conjecture, Sakai}

_[18]

_{considered the class} of chordal graphs. Sheshowed that

A(G)

< (A

+

3)2/4

for any chordalgraph

G.

For

a unit interval graph

G,

which is avery special chordal graph, she also proved that

2x(G)-

2

_<

A(G)

_<

2x(G).

Thepurposeofthis paperistostudy Griggsand Yeh’s conjectures.

We

alsostudy

L(2,

1)-labeling

numbersofthe union and the join oftwo graphstogeneralize results

on the n-wheelthat is thejoin of

_Cn

and

K1.

For

this purpose and afurther reason that will become clear in

3,

weintroducearelatedproblem,whichwecall the

U(2,

1)-labelingproblem. The definitionsofan

_{L’(2, 1)-labelin9}

f,

a

k-L’(2,

1)-labelin9

f,

and the

L’(2, 1)-labelin9

number

A’(G)

are the same as those ofan

_L(2,

1)-labeling f,

a

k-L(2,

1)-labeling f,

and the

L(2,

1)-labeling

number

A(G),

respectively, except that the function

_f

is required to be one-to-one. There is a natural connection between

A’(G)

and the path partition number

p(G

)

ofthe complement

G

of

G. For

any graph

G,

thepathpartition number

p(G)

is theminimumnumber k such that

V(G)

can bepartitioned intok paths.

The rest of this paper is organized asfollows. Section 2 gives general properties of

A(G)

and

A’(G).

Section 3 studies

_{A(GtOH), A(G+H),}

A’(GUH),

and

A’(G+H).

Section 4 proves that

A(G)

_< A

2

+

A

for a general graph

G

ofmaximum degree

A.

This result improves on Griggs and Yeh’s result

A(G)

<

A

2

₊

_2A.

_However,

there is stilla gap intheconjecture

A(G) <

A

2.

Section 5 studies the upperbounds for subclasses ofchordal graphs. Section 6 presents a polynomial time algorithm to determine

_A(T)

ofatree

T.

A

refereepoints out that

Georges,

Mauro,

and Whittlesey

[8]

alsosolved

A(G+H)

and

pv(G

+

H)

by a different approach. They actually gave the solutions without introducing the notionof

A’;

seethe remarks after

Lemmas

2.3 and 3.4.

2. Basic properties of

A

and

A

’.

<_

<_

_for

of

a

H.

LEMMA

2.2.

_A(G)

_<

_{A’(G) for}

anygraph

G.

_A(G)

_{A’(G) if}

G

is

_of

diameter at most two.

LEMMA

2.3.

pv(G)

A’(G

c)

-IV(G)I

+

2

_for

anygraph

G.

Proof.

Suppose

_f

is a

A’(GC)-L’(2,

1)-labeling

of

G

.

Note

that forany two vertices x and y in

V(G),

if

_f(x)

f(y)

₊

1, then

(x,y)

E(G)

and so

(x,y) e

E(G).

Consequently, asubsetofverticeswhoselabelsformaconsecutivesegment ofintegers form apath in

G.

However,

there are at most

A’(G)

-IV(G)I

+

2 such consecutive segments ofintegers. Thus p

(G) < A’(G)

IV

(G)]

+

2.

On

theother

hand,

suppose

V(G)

can be partitionedinto k

p(G)

paths in

G,

say,

(v,l,

v,2,...,

v,ni)

for 1

<

<

k. Consideradummy path

(v0,1)

and define

f

by

f(v,j)

f(v-.,n,_)

+

2,

f(v,j_)

+

1,

if 0 and j

1;

if 1

<

<

k and j 1; if 1

<

i

<

k and 2

<

j

<

ni.

It

isstraightforwardtocheckthat

_f

is a

_{(k+IV(G)I-2)-L’(2,}

1)-labeling

of

G

c.

Hence

A’(G)

<

k

₊

IV(G)I

2;

i.e.,

pv(G) >_

i’(G)

-IV(G)I

+

2. [3

Remark.

Georges,

Mauro,

and Whittlesey

[8,

Thm.

1.1]

provedthatforanygraph

G

ofn vertices the following twostatements hold.

(i)/(G) <

n- 1 if andonlyif

pv(G

)

1.

(ii)

Suppose

r is an integer greater than 1.

A(G)

n+r-2

if and only if

Note

thatan

_L(2,

1)-labeling

ispreciselyaproper vertexcoloringwithsomeextra conditionsonall vertex pairsofdistanceat most two.

So,

A(G)

hasanaturalrelation with the chromatic number

x(G).

For

any fixed positive integer

k,

thekthpowerofagraph

G

isthegraph

G

whose vertex set

V(G

)

V(G)

and edgeset

_E(G

k)

{(x,

y)"

1

da(x,

y)

k}.

LEMMA

2.4.

_x(G)

1

A(G)

2x(G

2)

2

_for

any graph

G.

Proof.

x(G)-

1

A(G)

follows from definitions.

A(G)

2x(G

2)

2 follows fromthefact that for any proper vertexcoloring

_f

of

G

,

_2f-

2 is an

L(2,

1)-labeling

of

G.

The neighborhood

N(x)

ofavertex xis thesetofall verticesy adjacent to x. The closed neighborhood

Nix]

ofxis

{x}

N(x).

LEMMA

2.5

_(see

_{[10]). A(G)}

A

₊

1

_for

any graph

G

_of

maximum degree

.

If

A(G)

A

₊

1,

then

f(v)

0 or

A

+

1

_for

any

A(G)-L(2,

1)-labeling

_f

and any vertex

v

_of

maximum degree

A.

In

this case,

Nix]

contains at most two vertices

_of

degree

A

for

any x

V(G).

LEMMA

2.6.

_{’(C3)= ’(C4)}

4 and

A’(C)

-n- 1

_for

n 5.

Proo

The cases of

_C3

and

C

are easy to verify.

For

n

5,

A(G)

n- 1 by definition.

Let

v0, Vl,...,vn-1 bevertices of

C

such that

v

is adjacent to V+l for

0 n-

1,

where

_v

v0. Consider the folIowing labeling:

f(v)

{

i/2,

if 0 n- 1 and is even;

[n/2

+

[i/2

-1,

if0in-landiisodd.

It

isstraightforwardtocheckthat

_f

is an

(n-

1)-L’(2,

1)-labeling

of

_Cn.

So

A’(C)

n-1.

H

LEMMA

2.7.

A’(P)

0,

A’(P2)

2,

A’(P3)

3,

and

A’(Pn)

-n- 1

_for

n 4.

Proof.

The casesof

P1,

P2, P3,

and

P4

areeasy toverify.

For

n

5,

A(P)

n- 1 bydefinition.

Last,

A’(P)

A’(C)

n-1 by

Lemmas

2.1 and 2.6.

H

3. Union andjoin ofgraphs.

Suppose G

and

H

aretwo graphswithdisjoint vertex sets. The unionof

G

and

H,

denoted by

G

H,

is the graph whose vertex set is

V(G) V(H)

and edge set is

E(G)

E(H).

The join of

G

and

H,

denoted by

G

₊

H,

is thegraph obtainedfrom

G

H

by adding all edges between vertices in

V

(G)

andvertices in

V(H).

LEMMA

3.1.

A(G

H)

max{A(G),

A(H)}

for

any two graphs

G

and

H.

Proof.

A(G

H)

max{A(G),

A(H)}

follows from

Lemma

2.1 and the fact that

G

and

H

are subgraphs of

G

H.

On

the other

hand,

an

_L(2,

1)-labeling

of

G

together with an

_L(2,

1)-labeling

of

H

makes an

_L(2,

1)-labeling

of

G

H.

Hence

H)

LEMMA

3.2.

A’(G

H)

max{A’(G), A’(H),

]V(G)[

+

]V(H)]-

1}

for

any two graphs

G

and

H.

Proo

A’(G H)

max{A’(G),

A’(H)}

followsfrom

Lemma

2.1 andthe factthat

G

and

H

aresubgraphs of

G

H.

A’(G

H)

V(G)I

+

V(H)-

1 followsfrom the definition of

.

312 GERARDJ. CHANG AND DAVID KUO

Assume

_f

is a

_{A’(G)-L’(2,}

1)-labeling

of

G.

Thereare notwo consecutive integers x

<

y in

_[0,

A’(G)]

that are not labels ofvertices of

G;

otherwise we can

"compact"

thefunction

_f

to get a

_{(A’(G)- 1)-L’(2,}

1)-labeling

_f’

of

G

defined by

f’(v)

{

f(v)’

if

f

(v) <

x;

f

(v)

l,

iff(v)>x.

For

the case where

A’(G)

> IV/)I

/

IVIH)I-

1,

here are at least

IVIH)I

pairwise

nonconsecutive integers in

[0,

A’(G)]

that are not labelsofvertices of

G.

We

canuse

them to label the vertices of

H.

This yields a

_{A’(G)-L’(2,}

1)-labeling

of

G

U

H.

For

thecase where

A’(H) >

IVIGDI

/

IVIHDI-

1, similarly, thereexists a

_{A’(H)-L’(2,}

1)-labeling of

G

U

H.

For

the case where

max{A’(G),A’(H)}

<

withoutlossof generality,wemay assumethat

]V(G)I

>

IV(H)].

Let

_f

bea

_k-L’(2,

1)-labeling of

G

such that k

<

IV/)I

/

IV(H)I-

and there are no two consecutive

integers in

[0, k]

that are not labels ofvertices of

G.

Such an

_f

exists for k

A’(G).

If

< IV()l

/

IV(H)I-

3,

then k

<

21v(G)l-

3 and sothere existtwo consecutive

labels x

<

y.

In

this case, we can

"separate"

_f

to get a

(k

+

1)-L’(2,

1)-labeling

_f’

defined by

f’

(v)

{

f(v),

if

f

(v) <

x;

f

(v)

+

l,

iff(v)>y.

Continuingthisprocess,weobtaina

_k-L’(2,

1)-labeling

such that

IV(G)]+IV(H)I-2

<

k

<

]V(G)I

+

IV(H)I-

1 and there are no two consecutive integers in

[0, k]

that are

not labels of vertices of

G.

Using

IV(H)]

nonlabels in

_{[0, ]V(G)I}

+

IV(H)I-

1]

to label the vertices in

H,

we get a

_(IV(G)]

+

_{IV(H)]- 1)-L’(2,}

1)-labeling

of

G

U

H.

By

theconclusionsofthe abovethree cases,

A’(GU

H) <

max{A’(G), A’(H), ]V(G)I

+

IV(H)I

1}.

n

LEMMA

3.3.

pv(G

U

_H)

pv(G)

+

pv(H)

for

any two graphs

G

and

H.

Proof.

Theproofisobvious.

LEMMA

3.4.

A(G

+

H)

A’(G

+

H)

A’(G)

+

A’(H)

+

2

_for

any two graphs

G

and

H.

Proof.

A(G

+

H)

A’(G

+

H)

follows from

Lemma

2.2 and thefactthat

G

₊

H

isofdiameter at most two.

Also,

pv((G

+

H)

c)

+

IV(G

+

H)I-

2

(by

Lemma

2.3)

p(G

U

H

)

+

IV(G)I

+

IV(H)I-p(G

)

+

,(H

)

+

IV(G)I

+

IV(H)I-

(by Lmm

3.3)

A’(G)+ A’(H)+

2

(by

Lemma

2.3).

El

Remark.

Georges,

Mauro,

and Whittlesey

[8,

Cor.

4.6]

proved that

A(G

+

H)

max{IV(G)l-

1,

A(G)}

+

max{IV(H)l-

1,/(H)}

+

2.

LEMMA

3.5.

p(G

+

H)

max{p(G) -IV(H)I,p(H)

-IV(G)I, 1} for

any two graphs

G

and

H.

Proof.

p,(C+H)

A’((G

+

H) ) -IV(G

+

H)I

+

2

(by

Lemma

2.3)

,’(G u

H

)

-IV(a)l-

IV(g)l

+

2

max{A’(G),A’(H), IV(G)I

/

IV(H)I- 1}

-Iv(G)I-

IV(H)I

+

2

(by

Lemma

3.2)

max{)’(G

)

-IV(a)l-4-

2-

IV(H)I,V(H

) -IV(H)I

+

2-

IV(a)l, 1}

max{pv(G)-

IV(H)I,p(H)-

Iv(G)I, x)

(by

Lemma

2.3).

El

Cographsare definedrecursively by the following rules.

(1)

A

vertexis a cograph.

(2)

If

G

is acograph, then sois its complement

G

c.

(3)

If

G

and

H

are cographs, thensois their union

G

U

H.

Note

that theabove definitionis thesame as onewith

(2)

replaced bythefollowing.

(4)

If

G

and

H

arecographs, then so istheir join

G

₊

H.

There isa linear timealgorithmtoidentifywhether agraph is acograph

(see

[3]).

In

the case ofa positive answer, the algorithm also gives a parsing tree.

Therefore,

we

have the followingconsequences.

THEOREM

3.6. There is a linear time algorithm to compute

A(G),

IV(G),

and pv

(G)

for

a cograph

G.

4.

Upper

bound of

A

interms ofmaximumdegree.

For

any fixed positive

integer

k,

ak-stablesetofagraph

G

isasubset

S

of

_V(G)

such thatevery two distinct

verticesin

S

areofdistancegreater than k.

Note

that 1-stabilityis the usualstability.

THEOREM

4.1.

_A(G)

_< A

2

+

A

for

any graph

G

withmaximum degree

A.

Proof.

Consider the following labelingschemeon

_V(G).

Initially, all verticesare unlabeled.

Let

S-1

.

When

_Si-1

is determined and not all vertices in

G

are

labeled,

let

F

{x

e

V(G)’x

is unlabeled and

d(x,

y) _>

2 for all y

e

i-1}.

Choose a maximal2-stable subset

_S

of Fi; i.e.,

S

isa 2-stablesubset of

_F

but

S

is

not aproper subset ofany 2-stable subset of

_F.

Note

thatinthecase where

Fi

,

i.e., for any unlabeledvertex x thereexistssomevertexy E

S-1

such that

d(x,

y)

<

2,

S

.

In

anycase, label allvertices in

S

byi. Then increase byone andcontinue

theaboveprocess untilallvertices are labeled.

Assume

kis themaximumlabel

used,

andchoose avertex x whose labelis k.

Let

II-{i’0ik-landd(x,y)=l

for

someyS},

I2={i’0ik-landd(x,y)2forsomeyS},

5

{i"

0 k-1 and

d(x,y)

3 for all y

S}.

It

isclear that

I2]

+

]I3]

k. Sincethe total number ofverticesywith1

d(x,

y)

2

is at most

deg(x)+

E{deg(y)-

(y,x)

E(G)}

A

₊

A(A-

)

A

:,

wehave

]I2[

A

2.

Also,

thereexist only

deg(x)

A

vertices adjacent to

x,

so

I[

A.

For

any

I3,

x Fi; otherwise

_Si

_{x}

is a 2-stable subset of Fi, which contradicts the choice of

_Si.

That is,

d(x,y)

1 for some vertex y in

Si-1;

i.e., i- 1

_I.

So,

5]

]21].

Then,

A(G)

k-

[I2[

+

[5[

[I2[

+

]I1[

2

₊

.

Jonas

[12]

proved that

A(G)

A

2

+

2A-

4 if

A(G)

k

2.

For

thecase of

A

3,

thisbound improves the boundin Theorem4.1 from 12 to 11.

5. Subclasses ofchordalgraphs.

A

graphis chordal

(or

triangulated)

ifevery cycle of length greater than three has a

chord,

which is an edge joining two non-consecutive vertices ofthecycle. Chordal graphs have been extensively studied as a subclass ofperfect graphs

(see

[9]).

For

any graph

G,

x(G)

denotes the chromatic number of

G

and

w(G)

the maximum size ofa clique in

G. It

is easy to see that

w(G)

x(G)

for anygraph

G.

A

graph

G

is perfectif

_w(H)

_x(H)

for any vertex-induced subgraph

H

of

G.

In

conjunction withthedominationtheoryingraphs, the followingsubclasses ofchordalgraphs have beenstudied

_(see

_[1,

2,

6]).

An

n-sunis a

chordal graphwith aHamiltoniancycle

(Xl,

Yl, x2, Y2,...,Xn, y,,

Xl)

inwhicheach

314 GERARDJ.CHANG AND DAVID KUO

xi is of degree exactly two.

An

SF-chordal

(resp.,

OSF-chordal,

3SF-chordal)

graph is a chordal which contains no n-sun with n

>_

3

(resp.

odd n

_>

3, n

3)

as an

induced subgraph. SF-chordal graphs are also called strongly chordal graphs by

Far-ber

(see

[6]).

Strongly chordal graphs include directedpath graphs, interval graphs,

unit intervalgraphs, block graphs, andtrees.

A

vertex x is simple if

N[y]

C_

N[z]

or

N[z]

C_

N[y]

for any two verticesy,z E

Nix].

Consequently,

for any simplevertex

x,

Nix]

is aclique and x has a maximum neighborrn

Nix];

i.e.,

N[y]

C_

N[m]

for any y

Nix].

Farber

[6]

proved that

G

is a strongly chordalgraph ifand only if every vertex-inducedsubgraph of

G

has asimple vertex.

THEOREM

5.1.

_/(G)

<_

2A

_for

any OSF-chordal graph

G

with maximum de-gree

A.

Proof.

First,

A(G)

<_

2x(G

2)

2 by

Lemma

2.4.

By

Corollary 3.11 of

[2],

G

2

is perfect and so

_x(G

2)

w(G2).

Since

G

is

OSF-chordal,

it is 3SF-chordal.

By

Theorem 3.8 of

[1],

w(G

2)

A

₊

1. The above inequality andequalities imply that

_<

THEOREM

5.2.

_/(G)

_<

/k-4-2X(G)-

2

_for

any strongly chordal graph

G

with

maximum degree

A.

Proof.

We

shall prove the theorem by induction on

_]V(G)I.

The theorem is

obviouswhen

IV(G)]

1.

Suppose

]V(G)I

>

1. Choose asimplevertexv of

G.

Since

G-

v is also strongly

chordal,

by the induction hypothesis,

Let

_f

be a

i(G- v)-L(2,

1)-labeling

of

G-

v.

Note

that v is adjacent to

deg(v)

vertices, whichformaclique in

G. Let

rn be the maximumneighbor ofv. Since every vertex ofdistance two from v is adjacent to m, there are

_deg(m)

_deg(v)

vertices that are of distance two from v.

Therefore,

there are at most 3

deg(v)

+

deg(m)

deg(v) _< A

+

2w(G)

2

A

₊

2x(G)

2 numbers used by

_f

to be avoided by v.

Hence

there is stillat least onenumber in

[0,

A

+

2x(G)

2]

that canbe assignedto

v inorderto extend

_f

into a

_(A

+

_2X(G)-

_2)-L(2,

1)-labeling.

D

Although

astrongly chordal graphis

OSF-chordal,

the upper bounds inTheorems 5.1 and 5.2 are incomparable. Theorem 5.2 is a generalization of the result that

A(T)

<_ A

₊

2 for any nontrivial tree ofmaximum degree

A. We

conjecture that

/(G)

_< A

₊

x(G)

for any strongly chordal graph

G

with maximumdegree

A.

6.

A

polynomial algorithm for

A

on trees.

For

a tree

T

with maximum

degree

A,

Griggs and Yeh

[10]

proved that

A(T)

A

₊

1 or

A

+

2. They also conjectured that it is NP-complete to determine if

A(T)

A

₊

1.

On

the contrary, this section givesapolynomialtimealgorithmto determine if

A(T)

A

/1.

Although

not necessary, thefollowing two preprocessing steps reduce the size of a treebefore

weapplythealgorithm.

First, checkifthere is a vertexx whose closed neighborhood

Nix]

contains three

ormoreverticesof degree

A.

Ifthe answeris positive, then

A(T)

A

₊

2 by

Lemma

2.5.

Next,

checkifthere is aleafx whose unique neighbor y hasdegree less than

A.

Ifthere is such a vertex

x,

then

T

x also hasmaximum degree

A. By

Lemma

2.1 and precisely thesame

arguments

asin theproof ofTheorem 4.1 of

[10],

A(T

x)

_<

(T)

_<

max{(T-x),

deg(x)+

2}

<_

_(T-x)and

so

_(T)=

_(T-x).

Determining

A(T)

is thenthesame as determining

A(T- x).

Continue this process until any leaf of the treeis adjacent to avertexof degree

A.

Regardless of whetherweapplythe above twostepstoreduce thetreesize or

not,

fromnow on weassume that

T

isatreeofat least two vertices and whose maximum degree is

A.

For

any fixed positive integer

k,

the following algorithm determines if

T

has a

k-L(2,

1)-labeling

or not.

We

in fact only need to apply the algorithm for

k=A+l.

For

technical reasons, we may assume that

T

is rooted at a leafr

,

which is

adjacenttor.

Let T

T

-

r berootedatr.

We

canconsider

T

asthetree resulting from

T

by adding a new vertex r that is adjacent tor only.

For

any vertexv in

T,

let

T(v)

be the subtreeof

T

rootedat v and

T’(v’)

bethe treeresulting from

T(v)

by addinga newvertex v thatisadjacenttovonly.

T(v

)

is considered to be rooted at theleaf

v’.

Note

that

T(r)

T

and

T’(r’)

T’.

Denote

S(T(v))

{(a, b)"

thereis a

k-L(2,

1)-labeling

_f

on

T’(v’)

with

f(v’)

a and

f(v) =b}.

Note

that

A(T)

5 k if and only if

_S(T(r))

=

O.

Now

suppose

T(v)-

v contains s

trees

T(vl),

T(v2),... ,T(vs)

rooted at Vl, v2,...,vs, respectively, where each

v

is adjacent tov in

T(v).

Note

that

T(v)

can be considered as identifying

v, v,...,

v,

to avertex v onthe disjointunionof

_T’(v{),

T’(v),..., T’(v’,).

For

asystem ofsets

_(A)=I

_(A,

A2,...,

A,),

asystem of distinct representa-tives

_(SDR)

is an s-tuple

(a)

*

_(a

_,..

i=1 a2

as)

ofs distinct elements such that

aEAforlhi<s.

THEOREM 6.1.

_S(T(v))

_{(a,b)"

0

<_

a

<_

k,

0

<

b

<_

k,

_la-bl

>_

2, and

(A)=

has an

SDR,

where

_A

{c

c

7

a and

(b,

c)

S(T(v))}}.

Proof.

Denote

by

S

theset onthe right-handsideof the equalityinthetheorem.

Suppose

(a,b)

S(T(v)).

There is a

k-(2,

1)-labeling

_f

of

T’(v’)

such that

f(v’)

a and

_f(v)

b. Of course, 0

<

a_<

k,

0

_<

b_<

k,

and

la-b]

_>

2.

Let

the same asv.

_Thenf

isa

f

be the function

_f

restricted on

_T’(v)

by viewing

_v

k-L(2,

1)-labeling

of

T’(v)

with

f(v)

f(v)

b and

f(v)

f(v)

=

f(v’)

a, i.e.,

(b, f(v))

e

S(T(v))

and

f(v)

e

A.

Thus

(f(v))=

isan

SDR

of

(A)*__.

This

proves

c_

s.

On

the other

hand,

suppose

(a,b)

S.

Then 0

<

a

<_

k,

0

<

b

<_

k,

_la-b

>_

2,

and

(A)%1

has an

SDR

(c)=.

Let

_f

be a

k-L(2,

1)-labeling

of

_T’(v)

such that

f(v)

b and

f(v)

c.

Consider the labeling

_f

of

T’

defined by

f(x)

f(x)

for x

V(T(v))

and

f(v’)

a.

It

is straightforward to confirm that

_f

is a

k-L(2,

1)-labeling of

T’(v’)

with

_f(v’)

a and

f(v)

b;

i.e.,

(a, b)

S(T(v)).

[:]

Our

algorithm for determiningifatree hasa

k-L(2,

1)-labeling

recursivelyapplies theabove theorem with the initial condition that for anyleafv of

T,

S(T(v))

{(a,b)"

0

_<

a

_<

k,

0

_<

b

_<

k,

[a-

b

>_

2}.

To

decide ifthetree

T’

has a

_k-L(2,

1)-labeling,

we calculate

S(T(v))

for all vertices v ofthe tree

T.

The algorithm starts from the leaves and works toward r.

For

any vertex

v,

whose childrenare

v,

v2,...,

v,

weuse

S(T(vl)),...

,S(T(v))

tocalculate

S(T(v))

by Theorem 6.1.

More

precisely, for any

(a,b)

with 0

<

a

<

k,

0

_<

b

_<

k,

la-bl

>_

2,

wecheck if

(a, b)

S(T(v))

bythefollowingmethod.

Construct

abipartite graph

G

(X, Y,

E)

with

X

{Xl,X2,...,Xs},

Y=

{0,

1,...,k},

E

_{(x,c)’c

a and

_(b,c)

e

S(T(v))}.

316 GERARDJ. CHANG ANDDAVIDKUO

We

can use anywell-known algorithm to find a maximum matching of the bipartite graph

G.

Then

(a,b)

E

S(T(v))

if and only if

G

has a matching ofsize s.

Note

that for any vertex vwe needto solve thebipartite matchingproblem

O(k

2)

times.

Therefore,

the complexity of the above algorithm is

O(IV(T)lk2g(2k)),

where

g(n)

is thecomplexity of solving the bipartite matching problem of n vertices. The well-known flow algorithmgives

g(n)

O(n2"5).

Acknowledgments. The authors wishto extend their gratitude tothe referee and to

Jerry

Griggs formany constructive suggestions for therevision ofthis paper.

REFERENCES

[1] G. J. CHANG AND (. L. NEMHAUSER (1984), The k-domination and k-stability problems on sun-freechordalgraphs,SIAMJ. Alg. Disc. Meth., 5,pp. 332-345.

[2] (1985), Covering, packing and generalized perfection, SIAM J. Alg. Disc. Meth., 6,

pp. 109-132.

[3] D. G. CORNEIL, Y. PERL, AND L. K. STEWART (1985), A linear recognition algorithmfor

cographs, SIAM J. Comput., 14,pp. 926-934.

[4] M. B. COZZENSANDF. S. ROBERTS (1982),T-Colorings_ofgraphs and the channel assignment

problem, Congr. Numer.,35,pp. 191-208.

[5] M. B. COZZENS AND D. I. WANG (1984), The general channel assignmentproblem, Congr. Numer.,41,pp. 115-129.

[6] M. FARBER (1983), Characterization_ofstrongly chordalgraphs, DiscreteMath., 43, pp. 173-189.

[7] Z. FOREDI, J. R. GRIGGS, AND D. J. KLEITMAN (1989), Pair labelings with given distance,

SIAMJ.Discrete Math., 2,pp. 491-499.

[8] J.GEORGES,D.MACRO,ANDM.WHITTLESEY(1994),Relatingpathcovering to vertexlabellings

witha conditionat distancetwo, DiscreteMath., 135,pp. 103-111.

[9] M. C. GOLUMBIC(1980), Algorithmic Graph Theoryand_PerfectGraphs,AcademicPress, New

York.

[10] J. R. GRIGGSAND R. K. YEH (1992), Labeling graphswithaconditionat distance2,SIAM J.

DiscreteMath., 5,pp. 586-595.

[11] W. K. HALE (1980), Frequency assignment: Theory and applications, in Proc. IEEE, 68,

pp. 1497-1514.

[12] K. JONAS (1993), Graph Coloring Analogues With a Condition at Distance Two:

L(2,1)-Labelings and List )-Labelings, Ph.D.thesis,Dept.ofMath.,Universityof SouthCarolina, Columbia, SC.

[13] A. RAYCHAUDHURI(1985), IntersectionAssignment, T-ColoringandPowers _ofGraphs,Ph.D. thesis,Dept.ofMath., RutgersUniversity, New Brunswick, NJ.

[14]

,

Further resultsonT-coloring andfrequency assignmentproblems,submitted.

[15] F. S. ROBERTS(1988), private communication toJ.R.Griggs.

[16] (1990), From garbage to rainbows: Generalizations_ofgraph colorings and their

appli-cations,inProc.of the Sixth InternationalConferenceontheTheoryandApplicationsof

Graphs, Y. Alavi, G. Chartrand, O.R.Oellermann,andA. J. Schwenk, eds., JohnWiley,

NewYork.

[17] (1991), T-colorings _ofgraphs: Recent results and openproblems, Discrete Math., 93,

pp. 229-245.

[18] D. SAKAI(1994), Labelingchordalgraphs: Distancetwocondition, SIAMJ.DiscreteMath., 7,

pp. 133-140.

[19] B. TESMAN (1989), T-Colorings, List T-Colorings, and Set T-Colorings of Graphs, Ph.D.

thesis,Dept.ofMath., RutgersUniversity, NewBrunswick, NJ.

[20] M. A. WHITTLESEY, J. P. GEORGES, ANDD. W. MACRO (1995), Onthe)-number_ofQn and

relatedgraphs,SIAMJ.DiscreteMath.,8,pp. 499-506.

[21] R. K. YEH (1990), Labeling Graphswitha Condition atDistance Two,Ph.D. thesis,Dept.of

Math., University ofSouth Carolina, Columbia, SC.