О совершенности графа C8,i,n Текст научной статьи по специальности «Науки о Земле и смежные экологические науки»

иркутским государственный университет путей сообщения

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Beijing : Science Press, 2008. P.5.

Er-gen Liu, Ke-wen Cai, Wu dan

UDC 519.17

ON THE GRACEFULNESS OF GRAPH C

8,7, n

1. Introduction

Posed by a number of circles graph is a kind of important and interesting graph, which gracefulness is on many scholars as the object of study [1~3]. The reference [4] is given graceful labeling of graph

1, 2n ' 2, 2n ' C6, 3, 2n . In this paper on the basis

of reference [4] to further expand, We proved that

graph Cgjiin, C8,2,n, Q8,3,nand Q8,4,n are the graceful graph.

In this paper, our discussion is undirected simple graph. V(G) and E(G) are vertex set and edge set of graph G . Unspecified symbols and terminology are the same reference [5].

Definition 1.1 For G = (V, E), If for each one v eV, there exists a non-negative integer 0(v) (called the vertices v of the label), meet:

(1) Vu, v eV, if u * v, then 0(u) * 0(v);

(2) max{0(v)| v e V} =| E |;

(3) V^, e2 e E,

if

* e2 =

then

0'(ex) * 0 (e2), which 0'(e) =| 0(w) - 0(v) |, e = uv.

Then G is called graceful graph. 0 is called graceful value or graceful labeling, 0' known as the edge of G induced value by 0.

Definition 1.2 Graph Cgin denotes a graph

that is composed by n loops Q8 which have i vertices adhered by order one after another, but from the second to 2n -1 circles , the adhered vertices, get 2 equidistant roads in each circle(not including the two circles adhered edge).

2. Main results

Theorem 2.1 Cg1n is a graceful graph.

Proof As the graph shown on fig. 1. Label all vertices as follows:

0(xu) = 0; 0(Xi2) = 8n; 0(x0) = 1; 0(xM) = 8n -1; 0(xi5) = 2 ; 0(x16) = 8n - 3 ; 0(x^) = 3; 0(x^) = 8n - 4;

0( x,-,2 ) = 3i + 4 + 8(n - i), i = 2,3, 0(x 3) = 3i, i = 2,3, —, n; 0(x; 4 ) = 3i + 5 + 8(n - i), i = 2,3, 0(x 5) = 3i -1, i = 2,3, —, n; 0( x ) = 3i + 2 + 8(n - i), i = 2,3,

, n;

, n;

в(хи1) = 37 +1, 7 = 2,3, —, n ; 0(x ) = 37 + 3 + 8(n - 7), 7 = 2,3, —,

, n;

n.

Now we proof that the follow 0 is graceful labeling of C81n

(1) Easy to find from the above label, In the same circle, these labels are different and the value of the label type contains n is greater than non- n value. In different circles, the value of the label type including n decreases with i increasing; the value of the label type excluding n increases with i increasing. Therefore, All the vertices have different labels and | E(CM,„) |= 8n = max0(x,. j), x;,^ e V(CSXn).

ft

* Fund Project: Natural Science Foundation of Jiangxi Province (0611009), Scientific research projects of Education Office (GJJ08254).

системным анализ и его приложения

X

11

X12 Xl3 Х14 X22 X23 X24

J V \

9 X

X

25

X18 X17 X16 X28 X27 X26

фи ,2 ¿n 39xn ,4

Xn ,8 Xn ,7 Xn,6

i=2 = fen,!

Fig. 1. The Graf C81,n.

(2) The derived value of edge 0' induced by 0 . e: |o(m)-0(v)||(u, v) = e g E(CSXn )} = {o( X,, j )-0(x, j+1)|| j = 1,2,-/7} U

{0(Xn) -0(Xi8)|}

U UUje(x,,j)-0(x,,j+i)||j = 2,3,-,7}

i=2

n

U U i 0(Xi,2 ) - 0(X,-1,5 ) |, | 0(X,,8 ) - 0(X,-1,5) |} =2

!,8n - 1,8n,-2,8n - 3,8n - 5,8n - 6,8n - 7} U i8n - 4}

[8(n - i) + 4,8(n - i) + 5,8(n - i) + 6,1 i=2 [8(n - i) + 3,8(n - i) + 1,8(n - i) + 2 J

n

U Ui8(n - i) + 8,8(n - i) + 7}

i=2

= {1,2,3,-,8n}.

Therefore 0' is the one-one correspondence from E(C8, 1> n ) to {1, 2,3, — ,8n}.

By (1), (2) we know that 0 is a graceful labeling of C8, 1 . Therefore C8,1,n is a graceful graph. Theorem 2.2 C82n is a graceful graph. Proof As the graph shown on fig. 2.

и U •

Label all vertices as follows: 0(xu) = 7n +1; 0(X12) = 1; 0(x13) = 7n ; 0(x14) = 2; 0(X15) = 7n - 2; 0(X16) = 3 ; 0(x^) = 7n - 3;

0( X18) = 0;

0( Xi,2) = •

в( Xi,3) = <

Xi,4) = •

Xi,5) = <

Xi,6) = <

в( Xi,7) = •

J 3i + 3 + 7(n - i), if n = 2k and 2 < i < n,

[3i, if i = 2k -1 and 2 < i < n;

[3i - 2, if i = 2k and 2 < i < n,

[3i +1 + 7(n - i), if i = 2k -1 and 2 < i < n;

[3i + 2 + 7(n - i), if i = 2k and 2 < i < n,

[3i -1, if i = 2k -1 and 2 < i < n;

[3i -1, if i = 2k and 2 < i < n,

[3i + 2 + 7(n - i), if i = 2k -1 and 2 < i < n;

[3i +1 + 7(n - i), if i = 2k and 2 < i < n,

[3i - 2, if i = 2k -1 and 2 < i < n;

[3i, if i = 2k and 2 < i < n,

13i + 3 + 7(n - i), if i = 2k -1 and 2 < i < n.

Now we proof that the follow 0 is graceful labeling of C81n

(1) Easy to find from the above label, In the same circle, these labels are different and the value of the label type contains n is greater than non- n value. In different circles, the value of the label type including n decreases with i increasing; the value of the

■JJi_23 ¿22 ¿23 щ,

T г ^-с

\ s X25 v

X

17

X16 X

27

26

Xn 2 _X -j

" ,2 a n ,3

X

n ,7

\ J

X

n ,4

X

n ,5

X

n ,6

Fig. 1. The Graf C8,2,n.

иркутским государственный университет путей сообщения

24

25

26

n ,3

' Xn-1 4 7 Xn,4

1 с s

n—1,5 n ,5

47

27

' n—1,6/~ Xn ,6

n ,7

Fig. 3. The Graf C^,n.

label type excluding n increases with i increasing. Therefore, All the vertices have different labels and

| E(C8,2,n ) |= 7n + 1 = maX 0<X, j ) , xi, j e V(C8,2,n ) . (2) The derived value of edge 0' induced by 0

0' : {0(u) -0(v)||(u, v) = e e E^)} = {0( x,, j)-0(x, j+1)|| j = 1,2, —,7} U {0( xu) -0( x18)|}

UU {0( x,, j)-0( x,, j+1)|| j = 2,3,—,6}

i=2

U JJ { 0( x,,2 ) - 0( x;-1,4 ) |, | 0(x,7 ) - 0(x-v) |}

i=2

= {7n,7n — 1,7n,—2,7n — 4,7n — 5,7 n — 6,7n — 3} U {7n +1}

U

U {7(n — i) + 5,7(n — i) + 4,7(n — i) + 3,7(n — i) + 2,7(n — i) +1}

i=2k 2<i<n

U U {7(n — i) + 7,7(n — i) + 6}

i=2k 2<i<n

U

U {7(n — i) + 1,7(n — i) + 2,7(n — i) + 3,7(n — i) + 4,7(n — i) + 5}

U U {7(n — i) + 6,7(n — i) + 7}

i=2k—1 2<i<n

i=2k—1 2<i<n

= {7n +1} U

.n. J7(n — i) + 1,7(n — i) + 2,7(n — i) + 3,7(n — i) + 4,] U |7(n — i) + 5,7(n — i) + 6,7(n — i) + 7 J

= {1,2,3, •■■,7n,7n +1}.

Therefore 0 is the one-one correspondence from E(Cgj2> J to {1, 2,3,,—,7n,7n +1}.

By (1), (2) we know that 0 is a graceful labeling ofC8 2 n. Therefore C82n is a graceful graph.

Theorem 2.3 C8 3 n is a graceful graph.

Proof As the graph shown on fig. 3. Label all vertices as follows:

0( xn) 0( xu) 0( xu) #(x18 ) = 6n — 1

0 ; в( xl2) = 6n + 2 ; 0( Xt3) = 1; 6n +1;

3 ; в(x^) = 6n ; в(x17) = 4 ;

в(x,3) =

3i + 2, if i = 2k and2 < i < n,

3i, if i = 2k — 1 and 2 < i < n;

3i + 3 + 6(n — i), if i = 2k and 2 < i < n,

3i + 4 + 6(n — i), if i = 2k — 1 and 2 < i < n;

в(x,4)

в(х^5 ) = 3i +1, i = 2,3,—, n

в( x,6)

3i + 4 + 6(n — i), if i = 2k and 2 < i < n, 3i + 3 + 6(n — i), if i = 2k — 1 and 2 < i < n;

в(x,7)=

[3i, if i = 2k and 2 < i < n, 13i + 2, if i = 2k — 1 and 2 < i < n.

Proof in imitation of Theorem 2.2.

(1) All the vertices have different labels and I E(Cgj3jB ) |= 6n + 2 = max 0(xi ] ) ,

Xi, J GV(Cg,3,n ) .

(2) 0 ' is the one-one correspondence from E(Cg,3, J to {1, 2,3, — ,6n,6n + 1,6n + 2}.

системным анализ и его приложения

By (1), (2) we know that 0 is a graceful labeling of C83n. Therefore C8 3 n is a graceful graph.

Theorem 2.4 CS4n is a graceful graph. Proof As the graph shown on fig. 4.

X

12

X

13

X

23

X

n—1,3 X

n ,3

X

X

X

^ ^ X14 ^ X24 J_J X15 J X2 5

lx lx

I X n-1,41 xn ,4 I Xn—1,5^ Xn ,5

17

X

16

X

26

X X

n—1,6 n ,6

Fig. 4. The Graf C8,4,n.

Label all vertices as follows: 0( xn) = 1; 0(X12) = 5n + 2; 0(x0) = 2; 0(xu) = 5n;

0(X15) = 3 ; 0(X16) = 5n -1; 0(X17) = 0; 0( X18) = 5n + 3;

[3i +1 + 5(n - i), if i = 2k and 2 < i < n, [3i -1, if i = 2k -1 and 2 < i < n; [3i, if i = 2k and 2 < i < n, |3i + 2 + 5(n - i), if i = 2k -1 and 2 < i < n; [3i + 2 + 5(n - i), if i = 2k and 2 < i < n, [3i, if i = 2k -1 and 2 < i < n; [3i -1, if i = 2k and 2 < i < n, 13i +1 + 5(n - i), if i = 2k -1 and 2 < i < n.

X,-3) = <

X,,4 ) = • *(X,,5 ) = <

d( X,6) = •

Proof in imitation of Theorem 2.2.

(1) All the vertices have different labels and

| E(C8 4,n) |= 5n + 3 = max 0(xi,j),

X, j (C8,4,„ ) .

(2) 0' is the one-one correspondence from E(CM>„) to {1, 2,3,—,5n,5n + 1,5n + 2,5n + 3}.

By (1), (2) we know that 0 is a graceful labeling ofC84n. Therefore C84n is a graceful graph.

Now we give the graceful labeling of C8,1,3 ,

C8,2,3 , C8,3,3 and C8,4,3 .

(1) The graceful labeling of C813 on fig. 5.

24 1 23 18 6 19 13 6 14

DDL)8

20 3 21 17 7 16 12 10 11

Fig. 5. The Graf C8,M.

(2) The graceful labeling of C8 2 3 on fig. 6.

1 21 16 4

8

18 3 6 14 12 Fig. 6. The Graf Cg^.

(3) The raceful labeling of C8 3 3 on fig. 7. 18 9

О

19 45 ► 13

3 » 7 ► 1О

18 М6 »12

4

Fig. 7. The Graf C8r3,3

6

11

(4) The graceful labeling of C8 4 3 on fig. 8. 17 _2 12 8

1 18

О

11 9

Fig. 8. The Graf C8,4,3.

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1. Frucht R. Graceful Numbering of Wheel and Related Graphs // [J]. Ann. N. Y. Acad. Sci. 1979. Vol. 319. № 219-229.

2. Dong Jun-Chao, Ma Mei-Jie. The Gracefulness of the Union of Some Circles // [J]. He bei Normal University. Natural Science Edition, 2000. Vol. 24 (1). P. 25-26.

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иркутским государственный университет путей сообщения

3. Siqinbater, Zhang Tian-Yu. On Gracefullness of Circles Union // [J]. Inner Mongolia University for Nationalities. Natural Science Edition, 2001. Vol. 16 (2). P. 113-114.

4. LIU Er-Gen, CAI Ke-Wen, WU Dan. On the Gracefulness of C6i2n // [J]. Journal of East Chi-

5.

na Jiaotong University. 2009. Vol. 26 (3). № 8184.

Ma Jieke. Graceful Graph // [M]. Beijing : Peking University Press, 1991.