The Vertex Independence Sequence of a Graph Is

The Vertex

Independence Sequence Is Not Constrained

of a

Graph

Tousef Alavi,

Paresh .I. Walde', and Allen .I. Sehwenk', Western Hichiqan University and Paul Erdds, Hungarian Academy of Sciences Abstract Ue conaider

ed with

indopbndont pendent

'r(2)

pereutatiom

that wmtex

G.

verticer

seto of order

determines er(lP

. sequ.ace

e fireph

of parametw.

lor

exuple,

in

G and eech Sorting

i.

a permutation

ei

.*,

tbia

a** , .*

Hat

stmciat-

maximmntir

ia then

w on the indicecr

into

the

of

number of inde-

aondecreasinR

order

m that

in UC?call a sequence constrained if certain S *+. r(a)' I cannot be realized by eey Kmph. It is well koova

the odge independence independence

show that,

y,

e fee be the

quite

every pereutetion

sequence

sequenca

the

contrary,

is reelized

ir

constrained

VP.B coniectured it

is totelly

to be uniwdol.

to be likevise,

but we

unconstreinod.

That ie,

The

by moae RrWh.

Constrained Sequences. We wish to study sequences of parameters associated with a specific graph. For example, in the vertex independence sequence 1.

al9 a2' -** , am, each ai denotes the number of independent sets of order i of vertices in G and m is the maximum order of any independent set. Similarly, in the edge independence sequence, bl, b2, *.* , b,, each bi counts the number of ways to select i independent edges. ARain m denotes the maximum order, but it is probably a different numerical value than m in the vertex indeThe approach we shall develop to study the vertex pendence sequence. independence sequence might well be applied to other graphical sequences, for example, to c3, ~4, **I , cm where ci counts the number of i-cycles in G. The edge independence sequence was shown.[l] is bl P.

bounds

each

are

m a-+ 1 of

30

not

eo far

K

II

unless 111 we have If

must

for

1.

permutation

Hi

ai

so,

have

m = 3

at and

from

the

-a*

, em

260

truth.

Determine

the

of order

sequence.

each

This

is

How-

independence independence

sequence. subKraph

Hi =

precisely

feature ei

a Specifical-

and no other

last

it an

sets

involved.

and construct

contains

is maximal,

if

to form

select

independence

Hi

only

it

independent

maximal

sequence

where

ei to

maximal

the

be-

of

without

ei Ii

the

.i graph

con-

affectinK

‘II

in

this

area.

The

first

two we list

above: smallest as the

order sorted

1arRe indices

enough

to realize

of

vertex

the

every

inde-

some Kraph.

Characterize

be realized?

we can even

to produce

remain

mentioned

of

is

the

order

sequence.

problems

been

that

but

i-set.

in

maximal

only

of which

to be selected the

counted

Not

maximal

each

giving

An independent

property

happens

specified

of

can be added

some contexts,

nonnegative

each

also

an additional

it

e2,

sequence.

is

to be a

in

sequence 2.

In because

independent

el,

of vertices

no vertex

+H +***+H 12 As constructed,*each

.

open

Ifany

in

to control.

, em this

sequence

sets

unconstrained,

terms

already

the

is,

context

easier

a maximal

pendence

< am

independence

that

‘I i-sets,

any other

if

independent

present

independent

Problem

wonder

+ l)-set.

G-H

- l)Ri

tions

to

to analyze

totally

pendence

mK 111

r = m - 1

by m

a

For

lower

counted

sequence

Problem

contains

to exceed

tends

overlap.

examples

best

is always

is

have

form

ordinary

sequence

allows

the

followed

of

suggests

be maximal,

are harder

tains

awl

to the

independent

fi

lots

m - 1 m-set

t-1

our

of maximal

similar

in

This

natural

haves

ever,

of

independent

Thus is

with

Problems

is

happens

Each

and there

Perhaps

It

starting

subsets,

We suspect

+ m.

for

permutations

the In

permutations

particular,

realized m-l can all 2

by the unimodal

edqe

inde-

pemuta-

It is possible is totally different

that for

Problem 3. For trees sequence unirodal?

the nature of the vertex trees or forests.

(or perhaps

forests?,

independence

is the vertex

sequence

independence

At one point we suspected that unimodality of G and H would imply G U Ii is unimodal. If so, the unimodal conjecture for trees would imply the one for forests. This is temptinK because it is easy to verify that settinE a0 - 1 by convention gives

%(G U H) -

t

ai(G)akei(H)

.

14

But such a convolution of unimodal sequences need not be uniaodal. For example, G = Kg5 + f7 has the u&modal sequence aG - 1, al - 116, = 147, whereas G U G has the nonunimodal sequence a3 * 343, a2 a0 = 1, al - 232, a2 * 13750, a3 - 34790, a4 = 101185, a5 - 100842, a6 - 117649. Thus, it would seem that the tree and forest conjectures need to be attacked separately. In closing, we suERest that permutation constraint offers a new perspective for investiqatinq other sequences associated with graphs. References 1. Allen .T. Schwenk, On unimodal sequences of graphical J. Goxbinatorial Theory B. 30(1981), 247-250.

ll

invariante,

nf

n,

n,

01

9

a3

123

0

27

11

87

1092

1331

132

0

38

9

103

1687

729

213

729

0

11

762

363

1331

231

1458

0

9

1485

243

729

312

729

38

1

806

1447

1

321

1458

27

1

1515

732

1

Tablet

For m-3 and T-36-729, themaximumorder - 1515 -12T +Zfi

21

+31

n

123 132

ni

0

at

a2

bj

4

12

48

64

2

7

n;,

rt;,

0

IO

12

8

213

37

0

4

49

48

64

231

53

0

4

65

48

64

312

0

3

1

9

12

I

0

1

4

3

1

321

1

Table 2, For m -3 , more carefulLy order - 65.

1234 1243 1324 1342 1423 1432 2134 2143 2314 2341 2413 2431 3r.24 3142 3214 3241 3412 3421 4123 4132 4213 4231 4312 4321

0 256 50 0 256 58 0 362 40 0 443 40 036258 0 443 50 65536 0 50 65536 0 58 131072 0 40 196608 040 13072 0 58 196608 050 65536 362 0 65536 443 0 131072 256 0 196608 256 0 I31072 443 0 196608 362 0 65536 362 58 65536 443 50 131072 256 58 196608 254 50 131072 443 40 196608 362 40

TabLe3.Form-4and mwmumoraer

21 IQ 21 19 I6 I6 21 19 21 19 18 ?a 21 19 21 IQ 16 16 1 1 1 1 1 1

chosen parameters

746 762 928 1082 962 1lOO 65770 65786 131276 196804 131310 196822 66344 66498 131668 197196 132022 197396 66436 66576 131762 197274 132082 197456

T-4865536, - 197456

22

75682 77794 138490 203215 142672 2U5285 10146 12258 7446 6966 11628 9036 133690 198415 68182 67702 197785 132580 141142 203755 75634 73042 20'1055 135850

grve maximum

182044 222548 101044 91436 211496 141384 162044 222548 101044 91436 211496 141384 37044 27436 37044 27436 16384 16384 195116 125004 195116 125004 64004 64004

194481 130321 194481 130321 65536 65536 194481 130321 194481 130321 65536 65536 194481 130321 194481 130321 65536 65536

Graph

IT

1234 1243 1324 1342 1423 1432 2134 2143 2314 2341 2413 2431 3124 3142 3214 3241 3412 3421 4123 4132 4213 4231 4312 4321

4 17

3KdJKs

I2

4K3

2KH + K+A,cR~ W2 l 4K5

4k X3 4K2 KS1+ Kffiu% l

k+

4K3

Km+K~u2K6 Kzer+K#.+2Kg Km+ 4K3 Kg7 + 4K3 K~+K,&.KB+K~UK~U~K~ Kz,+ 2KH* K3UKqu2Kg Km+ K++, + K&u2Kg Km+ K@C2,+ K&u2k Km+ 2Ku+ K+&u2k Kle6+K&*K&,u2& 3K, + 4K,

Table 4. Mm

KS% Kv* 4K2 KE + 4K2 K, l 4K, K3 * 4K,

45 64 8 14 lo8 55 296 301 82 IO9 296 296 297 301 301 302 16 5 25 33 5 7

M8 54 303 634 24 34

304 108 295 500 32 32

107 54 107 107 54 54 297 303 296 296 303 301 54 9 24 24 6 6

295 lo8 2% 295 108 108 295 295 295 295 295 295 68 7 32 32 4 4

320 81 300 625 16 16 300 81 300 300 81 81 300 300 300 300 300 300 1 2 16 16 I 1

careful chokes for m - 4 glve maximum order - 302.

23