A simple adrift graph G = with p vertices and q borders. A map degree Fahrenheit is called graceful labeling of graph G with Q borders if degree Fahrenheit is an injunction from vertices of G to the set { 0, 1, 2, 3, … , Q } such that when each border ultraviolet is assigned the label ensuing border labels are distinguishable.

A graph that admits graceful labeling is called graceful graph. The notation graceful labeling was introduced Rosa [ 6 ] with the name of rating.

Gallian [ 3 ] gives the extended study of parts to graceful labeling of assortment of graphs. The notation and nomenclature used in this paper are taken from [ 3 ] . There are many plants associating to graceful labeling of graphs with rhythms.

Rosa [ 6 ] showed that the n-cycle C is graceful if and merely if n 0 or 3 ( mod 4 ) . Wheels W = C + K1 are both graceful and harmonious [ 2 ] , [ 4 ] and [ 5 ] . The n-cone ( besides called the n-point suspension of Cm ) Cm + has been shown to be graceful when m 0 or 3 ( mod 12 ) by Bhat-Nayak and Selvam [ 1 ] . They besides prove that the undermentioned cones are graceful: C4 + , C5 + , C7 + , C9 + , C11 + and C19 + .

2. Main consequence.

Let C1, C2, . . . , Cn be n transcripts of rhythm Cm ( we term here as rings ) . Let the back uping points on the n rings C1, C2, . . . , Cn be t1, t2, … , Tennessee, which are merged severally to n dentitions of comb graph Pn a?ˆ L1. Let b1, b2, b3, … , bn be basal points of the comb from which n rings are hanging, each of which at a tooth of n dentitions severally. Let the points of ith ring be for one = 1, 2, … , N.

In this paper, the above construction is termed as Roman rings with n hanging rhythms suspended at thousand dentitions of comb severally and is denoted by R ( m, n ) graph and individual construction of the above is called section.

From the above definition of R ( m, n ) graph, it is clear that |V ( R ( m, n ) ) |= ( m+1 ) n. Besides figure of borders of R ( m, N ) is | E ( R ( m, n ) ) | = N ( m+2 ) -1

In this paper a graceful labeling of Roman rings R ( 3, N ) and R ( 4, N ) are discussed.

The entire figure of borders of R ( 3, N ) is given by q = 5n -1, where N denotes the figure of hanging rings.

For a particular type of graceful labeled, we assume n to be taken from the following tabular array.

N

J

R

N

J

R

3, 4, 5

n-1

1

27, 28, 29, 31

n-8

8

7, 8, 9

n-2

2

32,33

n-9

9

10,11

n-3

3

34, 35, 37, 38

n-10

10

13, 14, 15, 16

n-4

4

39, 40

n-11

11

17, 19

n-5

5

41, 43,44, 45

n-12

12

20, 21, 22, 23

n-6

6

46, 47

n-13

13

Continuing, the above for any n. In general for a brace n it is a?Y ( 2n/3 ) +1 and for group of four ? ( n/3 ) 2+1.

A general Roman ring of R ( 3, N ) drawn in the undermentioned Figure 1.

Figure 1

Throughout this paper, we besides denote the labeling of component of a given graph as cubic decimeter ( ? ) .

Here, for the Roman rings R ( 3, N ) , we assign the labeling with following regulations.

R ( 1 ) : cubic decimeter ( b1 ) = 0 ; cubic decimeter ( t1 ) = Q ( i.e. 5n-1 ) ; cubic decimeter ( b2 ) = q-2 ; cubic decimeter ( t2 ) = 2.

R ( 2 ) : cubic decimeter ( b2m+1 ) = cubic decimeter ( b2m-1 ) + 3, 1 ? m ? j. R ( 3 ) : cubic decimeter ( b2m +2 ) = cubic decimeter ( b2m ) -3, 1 ? m ? J.

R ( 4 ) : cubic decimeter ( t2m + 1 ) = cubic decimeter ( t2m-1 ) – 3, 1 ? m ? j. R ( 5 ) : cubic decimeter ( t2m + 2 ) = cubic decimeter ( t2m ) + 3, 1 ? m ? J.

It can be observe that cubic decimeter ( Bi ) + cubic decimeter ( Ti ) = Q.

R ( 7 ) : the labeling of cubic decimeter ( ) = cubic decimeter ( Bi ) + 1, for one = 1, 2 … , J.

R ( 8 ) : the labeling of cubic decimeter ( ) , one = 1, 2 … , J assigned in the undermentioned mode.

When J = n – R – 1, the labeling of vertices for the first J rings are q – 3j + 1in the descending order by 1 up to q – 4j + 1, for one = 1, 2, … , J.

If J is uneven so

cubic decimeter ( ) = q – 3j + 2.

cubic decimeter ( ) = cubic decimeter ( ) + 1, where s = 0, 1, … , ( j-3 ) /2.

Similarly cubic decimeter ( ) = 3j -1.

cubic decimeter ( ) = cubic decimeter ( ) – 1, where s = 1, … , ( j-3 ) /2.

If J is even so

cubic decimeter ( ) = 3j -1.

cubic decimeter ( ) = cubic decimeter ( ) – 1, where s = 0, 1, … , ( j-4 ) /2.

Similarly cubic decimeter ( ) = q – 3j + 2.

cubic decimeter ( ) = cubic decimeter ( ) + 1, where s = 1, … , ( j-2 ) /2.

The staying vertices of ( n – J ) rings of R ( 3, N ) are labeled as follows.

Let V1 ( G ) denote the set of values assigned to 4j vertices of the first J rings in the graceful labeling of R ( 3, n ) .

Let V2 ( G ) denotes the set of values ( { 0, 1, 2, … , Q } / V1 ( G ) ) from which the labeling of staying of 4 ( n – J ) vertices are to be labeled.

The values assigned to the vertices of V2 ( G ) are as follows.

( I ) . If J is even

Valuess runing from 3j/2 to q- ( 3j/2 ) except the values l ( ) for I = 1, 2, … , J.

( two ) . If J is uneven

Valuess runing from 3 ( ( j+1 ) /2 ) -1 to q- ( 3 ( j-1 ) /2 + 1 ) except the values l ( ) for I = 1, 2, … , J.

The labeling of first J sections of R ( 3, n ) bring on the graceful labeling of the first J sections of R ( 3, n ) .

Let E1 ( G ) denote the set of ( 5j – 1 ) values assigned to the borders of first J sections of R ( 3, n ) .

Let E2 ( G ) denotes ( { 1, 2, … , q } /E1 ( G ) ) be the values of the staying 5 ( n – J ) edges in the induced graceful labeling of vertices of R ( 3, n ) .

Let PV2 ( G ) and PE2 ( G ) be vertex set and border set PV2 ( G ) = { ? } , PE2 ( G ) = { ? } from which subsequently the values for the staying vertices and borders will be assigned.

If J is uneven so

( For computation of base vertex )

Measure 1: Lashkar-e-Taiba m* be maximum in E2 ( G ) , allow one = J.

Measure 2: l* = cubic decimeter ( Bi ) + m* .

Measure 3: If l* is non available in V2 ( G ) so

{ PE2 ( G ) = { PE2 ( G ) ? m* } , E2 ( G ) = { E2 ( G ) / { m* } } , go to step 1. }

( Note: the choice of m* starts from { m* to m*/2 } merely. )

Else

## {

I = i+1 ;

E2 ( G ) = { E2 ( G ) / { m* } } , V2 ( G ) = { V2 ( G ) / { l* } } and fifty ( Bi ) = l* . }

## ( For computation of dentitions vertex )

Measure 4: Lashkar-e-Taiba m* denote maximal in E2 ( G ) .

Measure 5: l* = l* – m* .

Measure 6: If l* is non available in V2 ( G ) so PE2 ( G ) = { PE2 ( G ) ? m* } , E2 ( G ) = { E2 ( G ) / { m* } } , go to step 4.

( Note: the choice of m* starts from m* to m*/2 merely. )

Else If l* is available in V2 ( G ) , so

E2 ( G ) = { E2 ( G ) / { m* } } , V2 ( G ) = { V2 ( G ) / { l* } } and fifty ( Ti ) = l* .

## ( For computation of ring )

Measure 7: Lashkar-e-Taiba m* be maximum in E2 ( G ) and allow M* be maximum in PE2 ( G ) .

Measure 8: if ( s = M* – m* ) available in E2 ( G ) so travel to step 10.

Else, step 9.

Measure 9: PE2 ( G ) = { PE2 ( G ) ? m* } , E2 ( G ) = { E2 ( G ) / { m* } } , go to step 7.

( Note: the choice of m* starts from m* to m*/2 merely. )

Measure 10: E2 ( G ) = { E2 ( G ) / { m* , s } } and E2 ( G ) = { E2 ( G ) ? { PE2 ( G ) / { M* } } } and PE2 ( G ) = { ? } .

Measure 11: Above divider relation is substituted to organize threes dwelling of cubic decimeter ( Ti ) , l ( ) and l ( ) ( their places can be swapped excessively ) , the values must be in V2 ( G ) .

( i.e. ) l* – M* = cubic decimeter ( ) and l ( ) ± m* = cubic decimeter ( ) .

Measure 11: If the threes are non available in V2 ( G ) so

PV2 ( G ) = { PV2 ( G ) ? l* } and PE2 ( G ) = { PE2 ( G ) ? m* } .

Travel to step 4.

On the completion of vertex values to ( j+1 ) Thursday section V2 ( G ) = { V2 ( G ) ? PV2 ( G ) } and E2 ( G ) = { E2 ( G ) ? unassigned values in PE2 ( G ) } and PV2 ( G ) = { ? } , PE2 ( G ) = { ? } .

The above procedure continued up to last ring.

If J is even so

( For computation of base vertex )

Measure 1: Lashkar-e-Taiba m* be maximum in E2 ( G ) , allow one = J.

Measure 2: l* = cubic decimeter ( Bi ) – m* .

Measure 3: If l* is non available in V2 ( G ) so

{ PE2 ( G ) = { PE2 ( G ) ? m* } , E2 ( G ) = { E2 ( G ) / { m* } } , go to step 1. }

( Note: the choice of m* starts from { m* to m*/2 } merely. )

Else

## {

I = i+1 ;

E2 ( G ) = { E2 ( G ) / { m* } } , V2 ( G ) = { V2 ( G ) / { l* } } and fifty ( Bi ) = l* . }

## ( For computation of dentitions vertex )

Measure 4: Lashkar-e-Taiba m* denote maximal in E2 ( G ) .

Measure 5: l* = l* + m* .

Measure 6: If l* is non available in V2 ( G ) so PE2 ( G ) = { PE2 ( G ) ? m* } , E2 ( G ) = { E2 ( G ) / { m* } } , go to step 4.

( Note: the choice of m* starts from m* to m*/2 merely. )

Else If l* is available in V2 ( G ) , so

E2 ( G ) = { E2 ( G ) / { m* } } , V2 ( G ) = { V2 ( G ) / { l* } } and fifty ( Ti ) = l* .

## ( For computation of ring )

Measure 7: Lashkar-e-Taiba m* be maximum in E2 ( G ) and allow M* be maximum in PE2 ( G ) .

Measure 8: if ( s = M* – m* ) available in E2 ( G ) so travel to step 10.

Else, step 9.

Measure 9: PE2 ( G ) = { PE2 ( G ) ? m* } , E2 ( G ) = { E2 ( G ) / { m* } } , go to step 7.

( Note: the choice of m* starts from m* to m*/2 merely. )

Measure 10: E2 ( G ) = { E2 ( G ) / { m* , s } } and E2 ( G ) = { E2 ( G ) ? { PE2 ( G ) / { M* } } } and PE2 ( G ) = { ? } .

Measure 11: Above divider relation is substituted to organize threes dwelling of cubic decimeter ( Ti ) , l ( ) and l ( ) ( their places can be swapped excessively ) , the values must be in V2 ( G ) .

( i.e. ) l* – M* = cubic decimeter ( ) and l ( ) ± m* = cubic decimeter ( ) .

Measure 11: If the threes are non available in V2 ( G ) so

PV2 ( G ) = { PV2 ( G ) ? l* } and PE2 ( G ) = { PE2 ( G ) ? m* } .

Travel to step 4.

On the completion of vertex values to ( j+1 ) Thursday section V2 ( G ) = { V2 ( G ) ? PV2 ( G ) } and E2 ( G ) = { E2 ( G ) ? unassigned values in PE2 ( G ) } and PV2 ( G ) = { ? } , PE2 ( G ) = { ? } .

The above procedure continued up to last ring.

The above set of delegating process continued up to last ring.

The general signifier of labeling of vertices as follows in Figure 2.

( missed border values ) Figure 2

For illustration, see R ( 3, 7 ) for general instance.

Figure 3

n= 7, J = 5, n – J = 2. The staying ( n – J ) rings, labeling of vertices and borders as follows.

In the above V2 ( G ) = { 8, 9, 10, 11, 12, 15, 16, 17, 18,19, 20, 24, 25, 26, 27 }

E2 ( G ) = { 15, 14, 11,10, 7, 6, 4, 3, 2, 1 }

m* in E2 ( G ) selected and in relation with cubic decimeter ( bj ) substituted such that the attendant value cubic decimeter ( bj+1 ) must be in V2 ( G ) . Again the m* in E2 ( G ) substituted for cubic decimeter ( tj+1 ) in relation with V2 ( G ) . The divider relation derived among borders such that the attendant vertex valleies in V2 ( G ) merely. Continuing we obtain the above graph.

Particular instance: if Q is premier ( 5 ( 6m ) – 1 ) it is non possible to label that n rings. While labeling of staying ( n – J ) rings in N, the border set contains more uneven Numberss instead than even Numberss.

Example: n= 6, 12, …

Example for particular instance Figure 4.

Figure 4

n = 6, J = 4, ( n – J ) = 2. After labeling of j rings the following are vertex set and border set.

V2 ( G ) = { 6, 7, 8, 9, 12, 13, 14, 15, 16, 17, 18, 21, 22, 23 }

E2 ( G ) = { 14, 13, 12, 11, 8, 7, 4, 3, 2, 1 } ; the even Numberss in E2 ( G ) are { 4, 8, 12, 14 } .

In add-on, the set V2 ( G ) contains fewer even Numberss in order to delegate uneven whole numbers.

We observe that by our process it is non possible to label above type of R ( 3, 6m ) . It leads to an interesting job of happening a graceful labeling of these particular instances.

3. Extension Roman Rings.

Suppose we may able to present rhythm C4 alternatively of C3, the entire figure of borders q = 6m-1.

For a particular type of graceful labeled, we assume n to fulfill the inequality 5r & A ; lt ; n ? 5 ( r+1 ) , where N is an whole number, R = 0, 1, 2, … and J = n-r-1.

The general signifier R ( 4, N ) given in the undermentioned Figure 5.

Figure 5

The labeling of vertices of R ( 4, n ) follows.

R ( 1 ) : cubic decimeter ( b1 ) = 0 ; cubic decimeter ( t1 ) = Q ; l ( ) = 1 ; cubic decimeter ( ) = 2 ; cubic decimeter ( b2 ) = q-3 ; cubic decimeter ( t2 ) = 3.

R ( 2 ) : cubic decimeter ( b2m+1 ) = cubic decimeter ( b2m-1 ) +4, 1 ? m ? j. R ( 3 ) : cubic decimeter ( b2m +2 ) = cubic decimeter ( b2m ) – 4, 1 ? m ? J.

R ( 4 ) : cubic decimeter ( t2m + 1 ) = cubic decimeter ( t2m-1 ) – 4, 1 ? m ? j. R ( 5 ) : cubic decimeter ( t2m + 2 ) = cubic decimeter ( t2m ) + 4, 1 ? m ? J.

It can be observe that cubic decimeter ( Bi ) + cubic decimeter ( Ti ) = Q.

Other vertex values of rings are labeled in following manner.

R ( 6 ) : the labeling of vertices l ( ) = cubic decimeter ( Si ) + 1, cubic decimeter ( ) = cubic decimeter ( Si ) +2 for one = 1, 2 … , J.

R ( 7 ) : the labeling of vertex values l ( ) assigned in the undermentioned status.

The ordered brace of border values are assigned from ( ( q+1 ) – 4j, q-4j ) in E ( ) and E ( ) of jth ring, following ordered brace of lower values in E ( ) and E ( ) of ( j-1 ) Thursday ring, traveling backwards continue up to first ring.

The first set of induced vertex and border values assigned up to j rings.

The staying border values are { 1, 2, 3, … , 6k } assigned in n – J rings in relation with lost vertex values assigned in n – J rings.

Let V2 ( G ) = { V ( G ) -V1 ( G ) } , E2 ( G ) = { E ( G ) – E1 ( G ) } . V1 ( G ) denotes the set of all aggregation of vertex points induced in the labeling and V2 ( G ) denotes the set of all vertex points yet to bring on for label ( i.e. ) the set contains 6n – J vertices. From the set 5 ( n – J ) vertices are labeled in the staying ( n – J ) rings and after labeling of full n rings ‘n ‘ vertices in ?1 remains.

V2 ( G ) contains the set of all vertices whose values runing from 4j/2 to q- ( 4j/2 ) demuring the values l ( ) for I = 1, 2, … , J is even.

V2 ( G ) contains the set of all values from 4 ( ( j+1 ) /2 ) -1 to q- ( 4 ( j-1 ) /2+1 ) demuring the values l ( ) for I = 1, 2, … , J is uneven.

The above labeling for first J sections of R ( 4, n ) bring on border values to the borders of the first J sections.

Let E1 ( G ) be the set of all border values induced due to ( n – J ) labeling of vertices in j sections of R ( 4, N ) and E2 ( G ) be the set of all losing border values yet to bring on. ( ( i.e. ) staying 6 ( n – J ) edges ) .

Let ?1 = { V2 ( G ) } ; ?2 = { E2 ( G ) } = { 6k, 6k-1, 6k-2, … , 4, 3, 2, 1 }

If J is even so

cubic decimeter ( bj+1 ) =l ( bj ) – upper limit in ?2 in E ( bjbj+1 ) relation with ?1. If non, following upper limit in ?2.

If J is uneven so

cubic decimeter ( bj+1 ) =l ( bj ) + maximal in ?2 in E ( bjbj+1 ) relation with ?1. If non, following upper limit in ?2.

If ( j+1 ) Thursday ring is uneven so

cubic decimeter ( tj+1 ) = six such that ei in E ( bj+1 tj+1 ) relation with cubic decimeter ( bj+1 ) .

i.e. cubic decimeter ( tj+1 ) = cubic decimeter ( bj+1 ) + ei, the value must be in ?1.

If ( j+1 ) Thursday ring is even so

cubic decimeter ( tj+1 ) = cubic decimeter ( bj+1 ) – ei, the value must be in ?1.

Now,

By ordered brace relation ( xi, Lolo ) and ( xj, yj ) eleven, Lolo, xj, yj. Choose any two brace substituted with cubic decimeter ( tj+1 ) , in relation the vertices cubic decimeter ( ) , l ( ) and l ( ) must be in ?1, continue up to last ring.

We observe that there are precisely n values in ?1 after labeling of vertices and borders.

The assignment of labeling of vertices given in the undermentioned Figure 6.

Figure 6

By utilizing the conditions the labeling of vertices in R ( 4, 5 ) in the undermentioned Figure 7.

Figure 7

n = 5, n – j=1 and j=4. The first four rings assigned ; the staying induced vertices and borders given below.

?1= { 8, 11, 12, 13, 14, 15, 16, 19, 20, 21 } ; ?2 = { 6, 5, 4, 3, 2, 1 }

the soap. in ?2 selected and in relation with cubic decimeter ( bn-k ) substituted such that the attendant value cubic decimeter ( bj+1 ) must be in ?1. Again the upper limit in ?2 substituted for cubic decimeter ( tj+1 ) in relation with ?1. The two divider dealingss derived among borders such that the attendant vertex valleies in ?1 merely. Continuing we obtain the above graph.