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Designs associated with maximum independent sets of cubic graphs

P.N. VINAY KUMAR1* and D. SONER NANDAPPA2

1Faculty of Mathematics, Government Science College, Hassan-573 201, Karnataka INDIA

2Department of Mathematics, University of Mysore, Mysore-570 006, Karnataka, INDIA

ABSTRACT

The independence number

( ) G

of a graph G is the maximum cardinality of an independent set of vertices in G. In this paper we study on partially balanced incomplete block designs associated with maximum independent sets of all cubic graphs on ten vertices.

Key words : Partially balanced incomplete block designs;

Maximum independent set; Independence number; Regular graphs.

Mathematics Subject Classification (2000): 05C

1. INTRODUCTION

Bose1 was the first to use graph t he ore t ic m e t hods t o s t udy t he partially balanced incomplete block designs (here after called PBIBD’s) a s s oc ia t e d wit h s t rongly re gula r graphs with two associate classes.

Harary et. al.8 have established the relation between isomorphic factoriza- tion of regular graphs and combina- torial designs. Ionin and Shrikande10

have studied some class of designs called

( , , , ) v k  

-designs over strongly regular graphs. Walikar et.al.12 have e sta blis hed t he rela t ion be t wee n minimum dominating sets of a graph with the blocks of PBIBD’s.

In this paper we obtain the PBIBD’s associated with maximum independent sets of all cubic graphs on ten vertices and we found that only five cubic graphs on ten vertices are

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associated with PBIBD’s.

2. Definitions and preliminary Results

Let

G  ( , ) V E

be a finite, conne- cted and undirected simple graph. A set of vertices in a graph G is indepen- dent if no two vertices in the set are adjacent. The independence number

( ) G

of a graph G is the maximum cardinality of an independent set of vertices in G. An independent set in G of cardinality

( ) G

is called a maximum independent set. For more results on maximum independent sets we refer Chang and Jou4. The terms used in this paper are used in the sense of Harary7. In this paper we establish a relation between maximum independent sets of cubic graphs on ten vertices with the blocks of PBIBD. In particular, the maximum independent sets of the Petersen graph form the blocks of PBIBD with parameters (10,5,2,4,0,1) with two association schemes.

Definition 2.1: A graph G is said to be a PBIB-graph, if the set of all maximum independent sets of G are the blocks of some PBIBD’s with m-association scheme.

Definition2.2[9]: Given symbols,

1, 2,..., v

, a relation satisfying the follo- wing conditions is called an m-class association scheme (

m  2

)

i. any two symbols are either 1st, 2nd

,… or mth associates; this relation being symmetric i.e., if the symbol

is the ith associate of then is the ith associate of .

ii. each symbol has ni i th associates, the number ni being independent of

.

iii. if and are two ith associates, then the number of symbols that are jth associates of and kth associates of is and

P

jki is independent of the pair of ith associate and .

Definition 2.3 [9]: Given  treatment symbols

1, 2,..., v

and an association of m-clas ses with

m  2

, we have a partially balanced incomplete block design (PBIBD) if treatment symbols can be arranged into b blocks each of them containing k symbols such that i. each of the symbol occurs in r blocks ii. every symbol occurs at most once in

a block.

iii. two symbols that are mutually ith associates occur together in exactly

i blocks..

The numbers

, , , , v b r k

i

( i  1, 2,..., ) m , , , , ( 1, 2,..., )

v b r k i m

are called the parameters of first kind, whereas the numbers

n

iand

P

jki ,

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, , 1, 2,...,

i j k

m are called the parame- ters of second kind.

Parametric relations2.4 [12]:

1.

vrbk

2.

1

1

m i i

n v

  

3.

1

( 1)

m i i i

n

r k

 

4.

1 m

i

jk j

i

P n

 

5.

n P

i jki

n P

j ikj

n P

k ijkw he r e , ,i j k1, 2,...,m , , 1, 2,...,

i j km.

Definition 2.5 [5]: The chro- matic number of a graph G is the smallest number of colors needed to assign the vertices of graph G and is denoted by

( )

G .

Observation 2.6: If G is a 3- regular graph on ten vertices then

( )

G

3.

3. RESULTS

In this section we prove that not all cubic graphs on ten vertices are PBIB-graph only five cubic graphs on ten vertices are PBIB-graph. We begin this section with a result on the bound f or indepe nde nc e num be r

( ) G

of a graph G which can found in Deo11.

Observation 3.1 [11]: If G is a gra ph w it h n ve rt ice s t he n,

( ) ( )

G n

G

 

  

 

 .

Lemma 3.2: For any cubic graph G on ten vertices

( ) G  4

.

Proof: From above observation

2.6 and 3.1, we have

10

( ) 4

G

3

    

. As G is cubic graph on ten vertices

( ) G 1

.To prove the le m ma it’s enough to prove that

( ) G  2

and

( ) G 3

. For if

( ) G  2

, let S

 { , }

u v be a maximum independent set. As G is a regular graph with each of the vertex of degree three, these two vertices covers only six vertices remaining four vertices are uncovered and from these we can choose one independent vertex. Thus, S is not an independent set and hence

( ) G  2

. For if,

( ) G  3

, let S

 { , , }

u v w be the maximum independent set. As G is cubic graph, the distance between each of the vertex in S must be three then only S covers all the vertices. But it is possible to choose a set of four vertices by choosing vertices which are at a distance two and which are inde- pendent and covers entire vertex set of G. Thus S is not maximum and hence.

( ) G 3

. Therefore

( ) G  4

. This

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proves the lemma.

Lemma 3.3: None of the cubic graphs shown in figure 1on ten vertices are PBIB-graphs.

Proof: We know that a graph is PBIB-graph if

vrbk

. To prove the result we consider only three graphs G1, G7and G11 and the remaining graphs can also be proved that they are not PBIB-graph in similar way. In G1, only one maximum independent set consisting of 3, where as there are three maximum independent sets containing 1, which violates the condition

vrbk

of PBIBD.

For G5, there are only three maximum independent set and only one of them consists of 4 and hence not a PBIB- graph. For the graph G11, there are eleven maximum independent sets are there and in that five maximum inde- pendent sets consists 5 and four maxi- mum independent sets consists 1, which violates

vrbk

of PBIBD and hence G11 is not a PBIB-graph.

The cubic graphs on ten vertices as shown in figure 2 are PBIB-graphs and we prove them in following results.

Proposition 3.4: The graph G17

is a PBIB-graph.

Proof: As we know that

( ) G  4

and each of the vertex of G17 belongs to exactly four maximum independent

sets. The following is the list of maxi- mum independent sets of G17.

{1,3, 5, 7},{1, 3, 5,8},{1, 4, 7, 9},{1, 5, 7, 9},{2, 4, 6, 9}

{2, 4, 6,10},{2, 4, 7, 9},{2, 6,8,10},{3, 5,8,10},{3, 6, 8,10}

Also, we observe from these sets that for any two distinct vertices of G17 exactly one of the following statements is true.

i.

v

1

and v

2 does not belong to any of the maximum independent set.

ii.

v

1

and v

2 belongs to exactly one off the maximum independent set.

iii.

v

1

and v

2 belongs to exactly two off the maximum independent sets.

iv.

v

1

and v

2belongs to exactly three off the maximum independent sets.

O n the ba s is of t he a bove properties, we define 4-association s che me a s follows : Two vert ice s

1

and

2

v v

are 1-associates if they does not belong to any of the maximum independent sets, 2-associates if they be long to exa ct ly one m axim um independent set, 3-associates if they be long to exa ct ly t wo m axim um independent sets and 4-associates if they belong to exactly three of the maximum independent sets of G17. With this definition, we give the table of association scheme as follows:

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Thus, the maximum independent sets of G17 with 4-association scheme defined above forms blocks of PBIBD with the parameters of first kind is given by,

1 2 3 4

10, 10, 4, 4, 0, 1, 2, 3

vbrk

1 2 3 4

10, 10, 4, 4, 0, 1, 2, 3

       

and the parameters of second

kind is given by

n

1

 3, n

2

 2, n

3

 2, n

4

 2

1

3,

2

2,

3

2,

4

2 nnnn

and

1

0 0 1 1

0 1 0 1

1 0 1 0

1 1 0 0

P

 

 

 

  

 

 

2

0 1 1 1

1 0 0 0

1 0 0 1

1 0 1 0

P

 

 

 

  

 

 

3

1 1 0 0

1 0 0 1

1 0 0 0

0 1 0 1

P

 

 

 

  

 

 

4

2 1 0 0

1 0 1 0

0 1 0 1

0 0 1 0

P

 

 

 

  

 

 

Vertices 1-associates 2-associates 3-associates 4-associates

1 2,6,10 4,8 3,9 5,7

2 1,3,5 7,8 9,10 4,6

3 2,4,9 6,7 1,10 5,8

4 3,5,8 1,10 6,7 2,9

5 2,4,6 9,10 7,8 1,3

6 1,5,7 3,9 4,8 2,10

7 6,8,10 2,3 4,5 1,9

8 4,7,9 1,2 5,6 3,10

9 3,8,10 5,6 1,2 4,7

10 1,7,9 4,5 2,3 6,8

Hence G17 is a PBIB-graph.

Proposition 3.5: The graph G

18

is a PBIB-graph.

Proof: The maximum indepen- dent sets of G

18 are given by

{1, 3,5, 7},{1, 3, 5, 9},{1, 3, 7, 9},{1, 5, 7, 9},{2, 4, 6, 8}

{2, 4, 6,10},{2, 4,8,10},{2, 6, 8,10},{3, 5, 7, 9},{4, 6,8,10}

We observe that each of the vertices belongs to exactly four maximum independent sets. We can easily verify

the conditions

vrbk

, as v

 10,

b

 10,

r

 4 and

k

 4 10, 10, 4 and 4

v

b

r

k

. By considering the maxi- mum independent sets above as a blocks and the 3-association scheme can be defined as the two distinct vertices v1

and

v2 are 1-associates if

1 2

( , ) 1

d v v

, 2-associates

d v v ( ,

1 2

)  2

and 3-associates if d v v

(

1

,

2

)  3

. With this definition we have the following table of association scheme:

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Verti- 1-asso- 2-asso- 3-asso- ces ciates ciates ciates

1 2,4,10 3,5,7,9 6,8

2 1,3,9 4,6,8,10 5,7

3 2,4,6 1,5,7,9 8,10

4 1,3,5 2,6,8,10 7,9

5 4,6,8 1,3,7,9 2,10

6 3,5,7 2,4,8,10 1,9

7 6,8,10 1,3,5,9 2,4

8 5,7,9 2,4,6,10 1,3

9 2,8,10 1,3,5,7 4,6

10 1,7,9 2,4,6,8 3,5

Thus with the above table of association scheme, G18 is a PBIB-graph with the parameters of first kind given by v

 10,

b

 10,

r

 4,

k

 4,

1

 0,

2

 3,

3

 0

1 2 3

10, 10, 4, 4, 0, 3, 0

      

and the parameters of second kind is given by,

n

1

 3, n

2

 4, n

3

 2

1

0 2 0 2 0 2 0 2 0 P

 

 

  

 

 

2

2 0 1 0 3 0 1 0 1 P

 

 

  

 

 

3

0 3 0 3 0 1 0 1 0 P

 

 

  

 

 

Hence G18 is a PBIB-graph.

Proposition 3.6: The Petersen graph G19 is a PBIB-graph.

Proof: The maximum independent sets of Petersen graph G19 is given below:

{1, 3, 6, 7},{1, 4,8,9},{2, 4, 6,10},{2, 5, 7,8},{3, 5, 9,10}

{1, 3, 6, 7},{1, 4,8,9},{2, 4, 6,10},{2, 5, 7,8},{3, 5, 9,10}

From this list of maximum inde- pendent sets we can see each of the vertex belongs to exactly two maximum independent sets and by considering the maximum independent sets listed above as blocks and the two associ- ation scheme can be defined as the two vertices

v

1

and v

2 are 1-associates if they are adjacent vertices in G19 and otherwise they are 2-associates. Thus with this give table of association scheme as follows:

Vertices 1-associates 2-associates

1 2,5,10 3,4,6,7,8,9

2 1,3,9 4,5,6,7,8,10

3 2,4,8 1,5,6,7,9,10

4 3,5,7 1,2,6,8,9,10

5 1,4,6 2,3,7,8,9,10

6 5,8,9 1,2,3,4,7,10

7 4,9,10 1,2,3,5,6,8

8 3,6,10 1,2,4,5,7,9

9 2,6,7 1,3,4,5,8,10

10 1,7,8 2,3,4,5,6,9

With this association scheme we can easily verify that the maximum independent sets of G19forms a PBIBD with t he paramete rs of firs t kind

1 2

10, 5, 2, 4, 0, 1

vbrk

and the parameters of the second kind is given by

n

1

 3, n

2

 6

and

1

0 2

2 4

P  

  

 

2

1 2

2 3

P

 

  

 

(7)

Hence the Petersen graph is a PBIB- graph.

Proposition3.7: The graph G20 is a PBIB-graph.

Proof: We can see that the following are maximum independent sets of the graph G20.

{1, 3, 6, 9},{1, 3, 7, 9},{1, 4, 6,8},{1, 4, 6, 9},{2, 4, 6,8}

{2, 4,8,10},{2, 5, 7,10},{2, 5, 8,10},{3, 5, 7, 9},{3, 5, 7,10}

From the list of maximum independent sets we can observe that each of the vertex belongs to exactly four of the maximum independent sets. Now, by using the 4-association scheme as defined in proposition 3.4, we obtain the following table of association scheme for G20.

Vertices 1-associates 2-associates 3-associates 4-associates

1 2,5,10 7,8 3,4 6,9

2 1,3,9 6,7 4,5 8,10

3 2,4,8 6,10 1,5 7,9

4 3,5,7 9,10 1,2 6,8

5 1,4,6 8,9 2,3 7,10

6 5,7,10 2,3 8,9 1,4

7 4,6,8 1,2 9,10 3,5

8 3,7,9 1,5 6,10 2,4

9 2,8,10 4,5 6,7 1,3

10 1,6,9 3,4 7,8 2,5

Thus, the maximum independent sets of G20 with 4-association scheme defined above forms blocks of PBIBD with the parameters of first kind is given by,

1 2 3 4

10, 10, 4, 4, 0, 1, 2, 3

v

b

r

k

1 2 3 4

10, 10, 4, 4, 0, 1, 2, 3

       

a nd t he para m e t e rs of

second kind is given by

n

1

 3, n

2

 2, n

3

 2, n

4

 2

1

3,

2

2,

3

2,

4

2 nnnn

and

1

0 0 1 1

0 1 0 1

1 0 1 0

1 1 0 0

P

 

 

 

  

 

 

2

0 1 1 1

1 0 0 0

1 0 0 1

1 0 1 0

P

 

 

 

  

 

 

3

1 1 1 0

1 0 0 1

1 0 0 0

0 1 0 1

P

 

 

 

  

 

 

4

2 1 0 0

1 0 1 0

0 1 0 1

0 0 1 0

P

 

 

 

  

 

 

Hence the graph G20 is a PBIB-graph.

Proposition 3.7: The graph G

21

is a PBIB-graph.

Proof: The maximum independent sets of G21are given by,

{1, 3,5, 7},{1, 3, 5, 9},{1, 3, 7, 9},{1, 5, 7, 9},{2, 4, 6, 8}

{2, 4, 6,10},{2, 4,8,10},{2, 6, 8,10},{3, 5, 7, 9},{4, 6,8,10}

T he as s oc ia t ion s c he m e on t he

(8)
(9)

vertices of G21is defined in the similar way as in the proposition 3.5. Thus the following table gives the 3-association scheme for the vertices of G21.

Ver- 1-asso 2-asso 3-asso tices ciates ciates ciates

1 2,6,10 3,5,7,9 4,8

2 1,3,7 4,6,8,10 5,9

3 2,4,8 1,5,7,9 6,10

4 3,5,9 2,6,8,10 1,7

5 4,6,10 1,3,7,9 2,8

6 1,5,7 2,4,8,10 3,9

7 2,6,8 1,3,5,9 4,10

8 3,7,9 2,4,6,10 1,5

9 4,8,10 1,3,5,7 2,6

10 1,5,9 2,4,6,8 3,7

Thus with the above table of association scheme, G21 is a PBIB-graph with the parameters of first kind given by v

 10,

b

 10,

r

 4,

k

 4,

1

 0,

2

 3,

3

 0

1 2 3

10, 10, 4, 4, 0, 3, 0

      

and the parameters of second

kind is given by,

n

1

 3, n

2

 4, n

3

 2

1

0 2 0 2 0 2 0 2 0 P

 

 

  

 

 

2

1 0 2 0 3 0 2 0 0 P

 

 

  

 

 

3

0 3 0 3 0 1 0 1 0 P

 

 

  

 

 

(10)

Thus, G21 is a PBIB-graph.

REFERENCES

1. Bose R.C. and Nair K.R., Partially balanced incomplete block designs, Sankhya, 4, 337-372 (1939).

2. Bose R.C., Strongly regular graphs, partial geometries and partially balanced designs, Pacific J. Math.

13, 389-419 (1963).

3. Bose R.C. and Shimamoto T., Classification and analysis of partially balanced incomplete block designs with two associate classes, J. Amer. Stat. Assn., 47, 151-184 (1952).

4. Chang G. J. and Jou M. J., The number of maximal independent sets in connected triangle-free graphs, Discrete Math., 197/198, 169-178 (1999).

5. Chartrand G. and Zhang P., Intro- duction to graph theory, Tata McGraw-hill Ic., New York (2006).

6. Cvetkovic D. M., Doob M. and Sachs H., Spectra of Graphs:

T he ory a nd applic ations ,

Academic Press, New

York (1980).

7. Harary F., Graph Theory, Addison- wesley, Reading, Mass (1969).

8. Harary F., Robinson R.W., Wormald N.C., Isomorphic factorization-I:

Complete graphs. Trans. Am. Math.

Soc. 242, 243-260 (1978).

9. Godsil C. and Royl G., Algebraic graph theory, Springer-Verlag, New York (2001).

10. Ionin Y.J. and Shrikande M.S., On classification of two class partially balanced designs, J. Stat. Plan.

Inform. 95, 209-228 (2001).

11. Narasingh Deo., Graph Theory-with applications to engineering and computer science, Prentice-Hall Inc., Englewood Cliffs, N.J., USA (2001).

12. Walikar H.B., Ramane H.S., Acharya B.D., Shekhareppa H.S., Arumugum S., Partially balanced incomplete block design arising from minimum dominating sets of paths and cycles, AKCE J. Graph. Combin.

4 (2), 223-232 (2007).

References

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