A Type of Arithmetic Labels about Circulating Ring

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A Type of Arithmetic Labels about Circulating Ring

Ergen Liu, Dan Wu, Kewen Cai

To cite this version:

Ergen Liu, Dan Wu, Kewen Cai. A Type of Arithmetic Labels about Circulating Ring. 4th Confer- ence on Computer and Computing Technologies in Agriculture (CCTA), Oct 2010, Nanchang, China.

pp.390-397, �10.1007/978-3-642-18369-0_45�. �hal-01564833�

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A type of Arithmetic Labels about Circulating ring Ergen Liu Dan Wu Kewen Cai

（School of Basic Sciences, East China Jiaotong University, Nanchang,P.R.China,330013）

E-mail:leg_eg@sina.com Absteact

The graph composed with several rings is a kind of important and interesting graph, many scholars studied on the gracefulness of this kind of the graph, The reference [1] is given the gracefulness of

m

kinds C₄ with one common point.

In this paper, we researched the arithmetic labels of four kinds graph: C₈_,₁_,_n^、C₈_,₂_,_n^、C₈_,₃_,_n^、C₈_,₄_,_n, and we proved they are all

( d , 2 d )

--arithmetic graph.

Key Word: Arithmetic graph；labeling of graph；C₈_,_i_,_n

1. Introduction

The graph in this paper discussed are undirected、no multiple edges and simple graph, the unorganized state of definitions and terminology and the symbols in this graph referred to reference[2][3].

There are two kinds of labels of the graph: one is the reduced label, is to say that in order to get the label of one edge you should reduce the endpoints of the edge; the other is additive label, for the same you should acttive the endpoints of one edge to get the label of the edge. For example, the well-known of “Gracefulness” is reduced.

the “Compatible labels” is additive. In 1990, B.D.Achaya and S.M.Hegde import the concept of “Arithmetic lables”(referred to reference[2]), which is a more extensive additive label, it have applied value on solution to question of the joint ventures in rights and obligations.

Definition 1.1 For G(V,E), if there is a mapping f ( called the vertices

v

of the label ) from V

 

G to the set of nonnegative integer N₀, meet:

) ( ) ( ) 1

( f u  f v

^，which

u  v

^，and u，vV(G)；



f(u) f(v)uv E(G)

 

k k d k (q 1)d



) 2

(    ， ，，  .

Then we call graph G is a (k，d)--arithmetic graph.

2. Main results and certification

Theorem 2.1 C₈_,₁_,_n is a (d,2d)--arithmetic graph.

Proof As the graph shown on fig.1.

) 0 (

v1 ) 0 (

v2

) 0 (

v3

) 0 (

v4

) 0 (

v5

) 0 (

v6

) 0 (

v7 ) 1 (

v1 ) 1 (

v2

) 1 (

v3

) 1 (

v4

) 1 (

v5

) 1 (

v6

) 1 (

v7 ) 2 (

v1 v⁽¹ⁿ^¹⁾

) 1 ( 2



vn

) 1 ( 3 n

v

) 1 ( 4



vn

) 1 ( 5 n

v

) 1 ( 6 n

v

) 1 ( 7 n

v

) ( 1

vn

Fig.1.The Graf C₈_,₁_,_n

(3)

Label all vertices as follows:

) , , 2 , 1 , 0 ( 8 )

(v₁⁽⁾ id i n

f ⁱ    , f(v₂⁽ⁱ⁾)8idd(i0,1,2,,n1), )

1 , , 2 , 1 , 0 ( 3 8 )

(v₃⁽⁾  id d i n

f ⁱ  , f(v₄⁽ⁱ⁾)8id6d(i0,1,2,,n1), )

1 , , 2 , 1 , 0 ( 2 8 )

(v₅⁽⁾  id d i n

f ⁱ  , f(v₆⁽ⁱ⁾)8id5d(i0,1,2,,n1), )

1 , , 2 , 1 , 0 ( 7 8 )

(v₇⁽⁾  id d i n

f ⁱ  .

Now we proof that the mapping

f

is arithmetic labeling of C₈_,₁_,_n.

We can see the mapping

f

meet f(u) f(v) which

u  v

and u,vV(C₈_,₁_,_n).

Next we prove



f(u) f(v)uvE(C₈_,₁_,_n)



is an arithmetic progression in the way of mathematical induction.

When n1

Then f(v₁⁽⁰⁾)0, f(v₂⁽⁰⁾)d, f(v₃⁽⁰⁾)3d, f(v⁽₄⁰⁾)6d, f(v₅⁽⁰⁾)2d, f(v₆⁽⁰⁾)5d, d

v

f( ₇⁽⁰⁾)7 , f(v₁⁽¹⁾)8d.

Therefore



f⁽u⁾ f⁽v⁾uvE⁽C8,1,1⁾

^



⁽ ⁾ ⁽ 2⁽⁰⁾^), )

0 (

1 f v

v

f 

 f(v₁⁽⁰⁾) f(v₃⁽⁰⁾), f(v₃⁽⁰⁾) f(v₅⁽⁰⁾), f(v₂⁽⁰⁾) f(v₄⁽⁰⁾), ),

( )

(v₅⁽⁰⁾ f v₇⁽⁰⁾

f  f(v₄⁽⁰⁾) f(v₆⁽⁰⁾), f(v₆⁽⁰⁾) f(v₁⁽¹⁾), f(v₇⁽⁰⁾) f(v₁⁽¹⁾)





d，3d，5d，7d，9d，11d，13d，15d



is an arithmetic progression, and the common difference is 2d. Suppose when nk1, we know



f⁽u⁾ f⁽v⁾uvE⁽C8,1,_k1⁾

^





d,d12d,d22d,,d(8k9)2d



is an arithmetic progression, and the common difference is 2d. Then when nk



f⁽u⁾ f⁽v⁾uvE⁽C8,1,k⁾

^



⁽ ⁾ ⁽ ⁾  ⁽ 8,1, 1⁾

^

 f u f v uv EC _k 



⁽ ⁾ ⁽ 2⁽ ¹⁾^), )

1 ( 1



  ^k

k f v

v

f f(v₁⁽^k^¹⁾) f(v₃⁽^k^¹⁾), f(v₃⁽^k^¹⁾) f(v₅⁽^k^¹⁾), ),

( )

(v₂⁽^k^¹⁾  f v⁽₄^k^¹⁾

f f(v₅⁽^k^¹⁾) f(v₇⁽^k^¹⁾), f(v₄⁽^k^¹⁾) f(v₆⁽^k^¹⁾), f(v₆⁽^k^¹⁾) f(v₁⁽^k⁾), f(v₇⁽^k^¹⁾) f(v₁⁽^k⁾)





d,d12d,d22d, ,d(8k9)2d

  



d(8k8)2d, d(8k7)2d, ,

2 ) 6 8

( k d

d   d(8k5)2d, d(8k4)2d, d(8k3)2d, d(8k2)2d,^d^⁽⁸^k^¹⁾^²^d





d d12d d22d d(8k1)2d

 ，，，，

is an arithmetic progression, and the common difference is 2 . d

In sum for the arbitrary nN₀, the mapping f:V(C₈_,₁_,_n)N₀ meet:

(1) f(u) f(v) when

u  v

and u,vV(C₈_,₁_,_n);

(2)



f(u) f(v)uvE(C₈_,₁_,_n)







d，d12d，d22d，，d(8n1)2d



. Therefore C₈_,₁_,_n is a (d,2d)--arithmetic graph.

Theorem 2.2 C₈_,₂_,_n is a (d,2d)--arithmetic graph.

(4)

) 0 (

v1

) 0 (

v2

) 0 (

v3

) 0 (

v4

) 0 (

v5

) 0 (

v6 ) 1 (

v1

) 1 (

v2

) 1 (

v3

) 1 (

v4

) 1 (

v5

) 1 (

v6 ) 2 (

v1

) 2 (

v2

) 1 ( 1



vn

) 1 ( 2



vn ) 1 ( 3



v n

) 1 ( 4



vn

) 1 ( 5



v n

) 1 ( 6



v n

) ( 1

vn

) ( 2

vn

Fig.2.The Graf C₈_,₂_,_n Label all vertices as follows:

) , , 2 , 1 , 0 ( 7 )

(v₁⁽⁾ id i n

f ⁱ    , )

, , 2 , 1 , 0 ( 7 )

(v₂⁽⁾ id d i n

f ⁱ     ,

) 1 , , 2 , 1 , 0 ( 3 7 )

(v₃⁽⁾  id d i n

f ⁱ  ,

) 1 , , 2 , 1 , 0 ( 6 7 )

(v⁽₄⁾  id d i n

f ⁱ  ,

) 1 , , 2 , 1 , 0 ( 2 7 )

(v₅⁽⁾  id d i n

f ⁱ  ,

) 1 , , 2 , 1 , 0 ( 5 7 )

(v₆⁽⁾  id d i n

f ⁱ  .

Now we proof that the mapping f is arithmetic labeling of C₈_,₂_,_n

We can see the mapping f meet f(u) f(v) which uv and u,vV(C₈_,₂_,_n).

Next we prove



f(u) f(v)uvE(C₈_,₂_,_n)



is an arithmetic progression in the way of mathematical induction.

When n1

Then f(v₁⁽⁰⁾)0, f(v₂⁽⁰⁾)d, f(v₃⁽⁰⁾)3d,f(v₄⁽⁰⁾)6d, f(v₅⁽⁰⁾)2d, f(v₆⁽⁰⁾)5d, d

v

f( ₁⁽¹⁾)7 , f(v₂⁽¹⁾)8d.

Therefore



f(u) f(v)uvE(C₈_,₂_,₁)





⁽ ⁾ ⁽ 2⁽⁰⁾^), )

0 (

1 f v

v

f 

 f(v₁⁽⁰⁾) f(v₃⁽⁰⁾), f(v₃⁽⁰⁾) f(v₅⁽⁰⁾), ),

( )

(v₂⁽⁰⁾ f v₄⁽⁰⁾

f  f(v₅⁽⁰⁾) f(v₁⁽¹⁾), f(v₄⁽⁰⁾) f(v₆⁽⁰⁾), f(v₆⁽⁰⁾) f(v₂⁽¹⁾), f(v₁⁽¹⁾) f(v₂⁽¹⁾)





d，3d，5d，7d，9d，11d，13d，15d



is an arithmetic progression, and the common difference is 2 . d Suppose when nk1, we know



f(u) f(v)uvE(C₈_,₂_,_k_₁)



＝



d,d12d,d22d,,d(7k7)2d



is an arithmetic progression, and the common difference is 2 . d Then when nk



f(u) f(v)uvE(C₈_,₂_,_k)





( ) ( )  ( ₈_,₂_, _₁)



 f u f v uv E C _k 



f(v₁⁽^k^¹⁾) f(v₃⁽^k^¹⁾), f(v₃⁽^k^¹⁾) f(v₅⁽^k^¹⁾), ),

( )

(v₂⁽^k^¹⁾  f v⁽₄^k^¹⁾

f f(v₅⁽^k^¹⁾) f(v₁⁽^k⁾), f(v₄⁽^k^¹⁾) f(v₆⁽^k^¹⁾), f(v₆⁽^k^¹⁾) f(v⁽₂^k⁾), f(v₁⁽^k⁾) f(v₂⁽^k⁾)





^d^,^d^¹^²^d^,^d^²^²^d^, ^d^⁽⁷^k^⁷⁾^²^d

  



^d^⁽⁷^k^⁶⁾^²^d^, d(7k5)2d, ,

2 ) 4 7

( k d

d   d(7k3)2d, d(7k2)2d, d(7k1)2d,d7k2d





d d12d d22d d7k2d

 ，，，，

is an arithmetic progression, and the common difference is 2 . d

In sum for the arbitrary nN₀, the mapping f :V(C₈_,₂_,_n)N₀ meet:

(5)

(1) f(u) f(v) when uv and u,vV(C₈_,₂_,_n);

(2)



f(u) f(v)uvE(C₈_,₂_,_n)







d，d 12d，d22d，，d7n2d



. Therefore C₈_,₂_,_n is a (d,2d)--arithmetic graph.

Theorem 2.3 C₈_,₃_,_n is a (d,2d)--arithmetic graph . Proof As the graph shown on fig.3.

) 0 (

v1 v2⁽⁰⁾

) 0 (

v3 v4⁽⁰⁾

) 1 (

v1 v2⁽¹⁾

) 1 (

v3 v4⁽¹⁾

) 2 (

v1 1⁽ ¹⁾



vn ⁽2 ¹⁾



vn

) 1 ( 3



v n ⁽4 ¹⁾



vn

) ( 1

vn

) 0 (

v0 v₀⁽¹⁾ v₀⁽²⁾ v⁽₀ⁿ^¹⁾ 0⁽ ⁾

vn

) 2 (

v3 3⁽ ⁾

vn

Fig.3.The Graf C₈_,₃_,_n Label all vertices as follows:

) , , 2 , 1 , 0 ( 8 )

(v₀⁽⁾ id i n

f ⁱ    , f(v₁⁽ⁱ⁾)4idd(i0,1,2,,n), )

, , 2 , 1 , 0 ( 3 4 )

(v₃⁽⁾ id d i n

f ⁱ     , f(v₂⁽ⁱ⁾)8id6d(i0,1,2,,n1), )

1 , , 2 , 1 , 0 ( 2 8 )

(v₄⁽⁾  id d i n

f ⁱ  .

Proof in imitation of Theorem 2.1.

For the arbitrary nN₀, the mapping f :V(C₈_,₃_,_n)N₀ meet:

(1) f(u) f(v) when uv and u,vV(C₈_,₃_,_n);

(2)



f(u) f(v)uvE(C₈_,₃_,_n)







d，d12d，d22d，，d(6n1)2d



.

By (1), (2) we know the mapping f is arithmetic labeling of

C

₈_,₃_,_n . Therefore C₈_,₃_,_n is a )

2 ,

(d d --arithmetic graph.

Theorem 2.4 C₈_,₄_,_n is a (d,2d)--arithmetic graph.

) 0 (

v1 ) 0 (

v2

) 0 (

v3

) 0 (

v4

) 1 (

v1 ) 1 (

v2

) 1 (

v3

) 1 (

v4

) 2 (

v1 v₁⁽ⁿ^¹⁾

) 1 ( 2



v n

) 1 ( 3



v n

) 1 ( 4



v n

) ( 1

v n

) 2 (

v3 v₃⁽ⁿ⁾

) 2 (

v2

) 2 (

v4

) ( 2

v n

) ( 4

v n

Fig.4.The Graf C₈_,₄_,_n Label all vertices as follows:

) , , 2 , 1 , 0 ( 5 )

(v₁⁽⁾ id i n

f ⁱ    , f(v₂⁽ⁱ⁾)5id d(i0,1,2,,n),

(6)

) , , 2 , 1 , 0 ( 3 5 )

(v₃⁽⁾ id d i n

f ⁱ     , f(v₄⁽ⁱ⁾)5id2d(i0,1,2,,n). Proof in imitation of Theorem 2.2.

For the arbitrary nN₀, the mapping f :V(C₈_,₄_,_n)N₀ meet:

(1) f(u) f(v) when uv and u,vV(C₈_,₄_,_n);

(2)



f⁽u⁾ f⁽v⁾uvE⁽C8,4,n⁾



=^



^d^，^d^¹^²^d^，^d^²^²^d^，^^，^d^⁽⁵ⁿ^²⁾^²^d



. By (1), (2) we know the mapping f is arithmetic labeling of C₈_,₄_,_n. Therefore C₈_,₄_,_n is a

) 2 ,

(d d --arithmetic graph.

3. Labels of Some Special Graph

In order to explan the correctness of the aforementioned labels, we give the arithmetic labeling of C₈_,₁_,₃

3 , 2 ,

C8 C₈_,₃_,₃ and C₈_,₄_,₃.

(1) The arithmetic labeling of C₈_,₁_,₃ on fig.5.

0 d

d 3 2d

d 6 5d

d 7

d 8

d 9

d 11

d 14

d 10

d 13

d 15

d 16

d 17

d 19 18d

d 22 21d

d 24 d

23 Fig.5. The Graf C₈_,₁_,₃

(2) The arithmetic labeling of C₈_,₂_,₃ on fig.6.

0

d d

3 2d

d

6 5d

d 7

d 8

d 9

d 14 d

10

d 13

d 15

d 16 d 17

d 19

d 22

d 21

d

12 20d

Fig.6. The Graf C₈_,₂_,₃ (3) The arithmetic labeling of C₈_,₃_,₃ on fig.7.

0 d

d

3 2d

d

6 5d

d 7 d 8

d 14

d 10

d 13

d 15 d

16

d 22 d 9

d

11 18d

d 24

Fig.7. The Graf C₈_,₃_,₃ (4) The arithmetic labeling of C₈_,₄_,₃ on fig.8.

(7)

0 d

d 3

d 2

d 6 d 5

d 7 d 8

d 10

d 13

d 15 d 16 d

11

d 18

d

12 17d

Fig.8. The Graf C₈_,₄_,₃

References

[1] Ma Kejie. Gracefulnss[M].Beijing:the press of technology,1991.

[2] Chartrand G,Lesniak L. Graphs and Digraphs[M].Wadsworth and Brooks\Cole,Monterey,1996.

[3] Bondy J A,Murty U S R. Graph Theory with Applications[M].Elsevier North-Holland, 1976.

[4] LIU Er-gen, WU Dan and CAI Ke-wen. Two Graphs Arithmetic Labeling[J], Journal of East China Jiaotong University. 2009, Vol.26(5): 89-92.

[5] Achaya B D and Hegde S M. Arithmetic graphs[J]. Journal of Graph Theory. 1990, 18(3):275-299.