Some Results on Prime and k − Prime Labeling

A graph G = (V, E) with n vertices is said to admit prime labeling if its vertices can be labeled with distinct positive integers not exceeding n such that the labels of each pair of adjacent vertices are relatively prime. A graph G which admits prime labeling is called a prime graph. In the present work we investigate some classes of graphs which admit prime labeling. We also introduce the concept of k−prime labeling and investigate some results concern to it. This work is a nice combination of graph theory and elementary number theory.


Introduction
We begin with simple, finite, undirected and non-trivial graph G = (V, E) with the vertex set V and the edge set E. The number of elements of V, denoted as |V| is called the order of the graph G while the number of elements of E, denoted as |E| is called the size of the graph G.In the present work C n denotes the cycle with n vertices and P n denotes the path of n vertices.In the wheel W n = C n + K 1 the vertex corresponding to K 1 is called the apex vertex and the vertices corresponding to C n are called the rim vertices.The graph f n = P n−1 + K 1 is called a fan and the vertex corresponding to K 1 is called the apex vertex of the fan.For various graph theoretic notations and terminology we follow (Gross, J. & Yellen, J., 2004) whereas for number theory we follow (Burton, D. M.,1990).We will give brief summary of definitions and other information which are useful for the present investigations.Definition 1.1 If the vertices of the graph are assigned values subject to certain conditions then it is known as graph labeling.
For latest survey on graph labeling we refer to (Gallian, J. A., 2009).Vast amount of literature is available on different types of graph labeling and more than 1000 research papers have been published so far in last four decades.For any graph labeling problem following three features are really noteworthy: • a set of numbers from which vertex labels are chosen; • a rule that assigns a value to each edge; • a condition that these values must satisfy.
The present work is aimed to discuss one such labeling known as prime labeling.Definition 1.2 A prime labeling of a graph G is an injective function f : V → {1, 2, • • • , |V|} such that for every pair of adjacent vertices u and v, gcd( f (u), f (v)) = 1.The graph which admits prime labeling is called a prime graph.
The notion of prime labeling was originated by Entringer and was discussed in (Tout, A., 1982, p. 365-368).Many researchers have studied prime graphs.It has been proved by (Fu, H. L., 1994, p. 181-186) P n on n vertices is a prime graph.It has been proved by (Lee, S. M., 1988, p. 59-67) wheel graph W n is a prime graph if and only if n is even.In (Deretsky, T. D., 1991, p. 359-369) cycle C n is a prime graph.Definition 1. 3 The graph G = < W n : W m > is the graph obtained by joining apex vertices of wheels W n and W m to a new vertex w.Definition 1.4 A t-ply P t (u, v) is a graph with t paths, each of length at least two and such that no two paths have a vertex in common except for the end vertices u and v. Definition 1.5 Two prime integers are said to be twin primes if they differ by 2.
Thus in each of the possibilities the graph G under consideration admits a prime labeling.Which implies that G is a prime graph.
Illustration 2.8 A prime labeling of the graph defined by identifying an end vertex of P 6 with the apex vertex of W 8 is shown in the Figure 4. Theorem 2.9 Let G 1 be a prime graph of order n 1 with a prime labeling f and having vertices u 1 and u n 1 with the labels 1 and n 1 respectively.Then the graph G obtained by identifying an end vertex of a path P n 2 with either u 1 or u n 1 of G 1 is a prime graph.
Proof: Let the vertices of a prime graph G 1 be u 1 , u 2 , • • • , u n 1 and the prime labeling f of G 1 be such that f (u 1 ) = 1 and f (u n 1 ) = n 1 .Let v 1 , v 2 , • • • , v n 2 be the consecutive vertices of P n 2 .We have the following two cases: 1.The graph G is obtained by identifying an end vertex v 1 of P n 2 with the vertex u n 1 of G 1 .(The proof is similar if the other end vertex v n 2 of P n 2 is identified with the vertex u n 1 of G 1 .)Define a labeling function g on G as follows: Obviously g is an injection.Also g is an extension of the prime labeling function f on G, it is enough to prove the following cases: (a) gcd(g(u n 1 ), g(v 2 )) = 1.To prove this we have gcd(g(u n 1 ), g(v 2 )) = gcd( f (u n 1 ), g(v 2 )) = gcd(n 1 , n 1 + 1) = 1 as n 1 and n 1 + 1 are consecutive integers.(b) For each j ∈ {2, 3, • • • , n 2 − 1}, gcd(g(v j ), g(v j+1 )) = 1.To prove this we have gcd(g(v j ), g(v j+1 )) = gcd(n 1 + j − 1, n 1 + j) = 1 as n 1 + j − 1 and n 1 + j are consecutive integers.
2. The graph G is obtained by identifying the other end vertex v 1 of P n 2 with the vertex u 1 of G 1 .(The proof is similar if the other end vertex v n 2 of P n 2 is identified with the vertex u 1 of G 1 .) Define a labeling function g on G as follows: Obviously g is an injection.Also g is an extension of the prime labeling function f on G, it is enough to prove the following cases: (a) gcd(g(u 1 ), g(v 2 )) = 1.To prove this we have gcd(g(u 1 ), g(v 2 )) = gcd( f (u 1 ), g(v 2 )) = gcd(1, n 1 + 1) = 1.(b) For each j ∈ {2, 3, • • • , n 2 − 1} we need to show that gcd(g(v j ), g(v j+1 )) = 1.To prove this we have gcd(g(v j ), g(v j+1 )) = gcd(n 1 + j − 1, n 1 + j) = 1 as n 1 + j − 1 and n 1 + j are consecutive integers.
Thus in each of the possibilities the graph G under consideration admits a prime labeling.Which implies that G is a prime graph.
Theorem 2.10 A graph G obtained by identifying all the apex vertices of m fans f n 1 , f n 2 , • • • , f n m ( is called a multiple shell) is a prime graph.
Proof: A fan graph f n = P n−1 + K 1 has n vertices and 2n − 3 edges.Let the graph G is obtained by fusing all the apex vertices of f n 1 , f n 2 , • • • , f n m .Let the common apex vertex of each of the fans f n i after fusing all the apex vertices of all the fans f n 1 , f n 2 , • • • , f n m be v 0 .For each i ∈ {1, 2, 3, • • • , m}, denote the remaining vertices of the fan Here we define y x a = 0 if x and y are any positive integers with y < x.First we will show that f is an injection.It is easy to check that f is a prime labeling.Let e = uv be an edge of G. Then clearly it must be an edge of exactly one of the fans f n i for some as consecutive integers are relatively prime.Thus f admits a prime labeling for G. i.e.G is a prime graph.
Published by Canadian Center of Science and Education Illustration 2.11 The graph G obtained by identifying all the apex vertices of three fans f 3 , f 4 , f 5 has prime labeling is shown in Figure 5.
Theorem 2.12 A graph G obtained by identifying all the apex vertices of m wheels Proof: Let the common apex vertex of G be u 0 and the consecutive rim vertices of each of the wheels Here we define y x a = 0 if x and y are any positive integers with y < x.
To prove f is injective it is enough to prove that f is surjective as the cardinality of the domain and codomain are same.
For each y ∈ {1, 2, It shows that f is surjective.Let e = xy be an edge of G then it must be an edge of one of the wheels W n i for some i ∈ {1, 2, • • • , m}. we have the following two possibilities: 1.If one of the end vertices of e is the apex vertex u 0 with x = u 0 then gcd ( 2. If none of the end vertices of e is the apex vertex u 0 and n k are consecutive integers so they necessarily be relatively prime.
Which shows that f admits a prime labeling i.e.G is a prime graph.

Illustration 2.13
The graph G obtained by identifying the apex vertices of two wheels W 6 and W 8 has prime labeling is shown in Figure 6.
Theorem 2.14 A t-ply graph P t (u, v) is a prime graph if the order of P t (u, v) is a prime number.
Proof: Suppose a t-ply P(u, v) is obtained from t distinct paths P i , for each i = 1, 2, • • • , t, each of length n i , such that the vertices of The number of vertices of P t (u, v) is a prime number p with where p is a prime number.Here we define y x a = 0 if x and y are any positive integers with y < x.First we will show that f is an injection.It is easy to check that f (w) = 1 if and only if w = u as well as f (w) = p if and only if w = v.Suppose f (v i, j ) = f (v i , j ) for some positive integers i, j, i , If i i then without loss of generality assume that i < i , so i ≤ i − 1. Then It is enough to show that gcd( f (x), f (y)) = 1, for every pair of adjacent vertices x and y.Let e = xy be an edge of G. Then clearly it must be an edge of exactly one of the paths P n i for some i = 1, 2, • • • , t.
1.If one of the end vertices of e is u say x = u then gcd( 2. If one of the end vertices of e is v say 3. If none of the end vertices of e is u and v then clearly {u, v} = {v i, j , v i, j+1 } for some j ∈ {1, 2, as any two consecutive integers are relatively prime.
Thus f admits a prime labeling for G.That is G is a prime graph.
Illustration 2.15 The 5-ply graph obtained by taking five paths of lengths 6, 6, 4, 6 and 10 respectively has prime labeling is shown in Figure 7.
The graph which admits a k−prime labeling is called a k−prime graph.
One note that every prime graph is a k−prime graph for k = 1.
Lemma 3.2 For each positive integer m the path graph P m is a k−prime graph for each positive integer k.
Proof: Denote the vertices of Thus the path graph P m is a k−prime graph.

Theorem 3.3
The graph G obtained by disjoint union of a prime graph G 1 of order n 1 and a (n 1 + 1)−prime graph G 2 is a prime graph.
Let the graph G be obtained by disjoint union of G 1 and G 2 .
Obviously f is an injection.Let e = xy be an arbitrary edge of G. Then either e ∈ E(G 1 ) or e ∈ E(G 2 ).
Thus f admits a prime labeling of G and consequently G is a prime graph.
Theorem 3.4 Let G 1 be a prime graph of order n 1 with a prime labeling f 1 and having vertices u 1 and u n 1 with f 1 (u 1 ) = 1 and f 1 (u n 1 ) = n 1 .Let G 2 be a n 1 −prime graph of order n 2 with a n 1 -prime labeling f 2 having a vertex v 1 with f 2 (v 1 ) = n 1 .
Then the graph G obtained by identifying the vertex v 1 of G 2 with either to u 1 or to u n 1 of G 1 is a prime graph.
We claim that f is an injection because Let e = xy be an arbitrary edge of G. Then either e ∈ E(G 1 ) or e ∈ E(G 2 ). (a 2. Consider the graph G obtained by identifying the vertex We claim that f is an injection because Let e = xy be an arbitrary edge of G.
Thus in each of the possibilities f admits a prime labeling of G consequently G is a prime graph.
Corollary 3.5 A tadpole (graph obtained by identifying a vertex of a cycle to an end vertex of a path) is a prime graph.
Proof: As we know that every cycle is a prime graph.According to Lemma 3.2 every path is k−prime graph for every positive integer k.Then using Theorem 3.4 a tadpole is a prime graph.

Conclusion
Here we investigate eight results corresponding to prime labeling.We introduce a new concept of k−prime labeling and derive four results.Analogous work can be carried out for other families and in the context of different types of graph labeling techniques.We begin with simple, finite, undirected and non-trivial graph G = (V, E), with vertex set V and edge set E. In the present work C n denote the cycle with n vertices and P n denote the path of n vertices.In the wheel W n = C n + K 1 the vertex corresponding to K 1 is called the apex vertex and the vertices corresponding to C n are called the rim vertices where n 3. Throughout this paper |V | and |E| are used for cardinality of vertex set and edge set respectively.We assume F 1 = 1, F 2 = 2 and for each positive integer n, F n+2 = F n+1 + F n .For each positive integer n, F n is called the nth Fibonacci number.For various graph theoretic notations and terminology we follow Gross and Yellen [3] while for number theory we follow Burton [1].We will give brief summary of definitions and other information which are useful for the present investigations.
Definition 1.1 If the vertices of the graph are assigned values subject to certain conditions then it is known as graph labeling.
Vast amount of literature is available in printed as well as in electronic form on different types of graph labeling.More than 1200 research papers have been published so far in last four decades.Most interesting graph labeling problems have following three important ingredients.
• a set of numbers from which vertex labels are chosen; 1 Received April 15, 2011.Accepted November 24, 2011.
• a rule that assigns a value to each edge; • a condition that these values must satisfy.
Most of the graph labeling techniques trace their origin to graceful labeling introduced by Rosa [5].
The problem of characterizing all graceful graphs and the graceful tree conjecture provided the reason for different ways of labeling of graphs.Some variations of graceful labeling are also introduced recently such as edge graceful labeling, Fibonacci graceful labeling, odd graceful labeling.For a detailed survey on graph labeling we refer to Gallian [2].The present work is aimed to discuss Fibonacci graceful labeling.
The notion of a Fibonacci graceful labeling was originated by Kathiresan and Amutha [4].They have proved that K n is Fibonacci graceful if and only if n 3 and path P n is Fibonacci graceful.13 If a graph G admits a super Fibonacci graceful labeling then G is called a super Fibonacci graceful graph.
With reference to Definitions 1.3 and 1.5 we observe that in any (super) Fibonacci graceful Then the graph obtained by joining a vertex of G i to the corresponding vertex of G i+1 by an edge for Motivated through this definition we define the following.
In the next section we investigate some new results on Fibonacci graceful graphs.§2.Some results on Fibonacci Graceful Graphs Theorem 2.1 The graph obtained by joining a vertex of C 3m and a vertex of C 3n by an edge admits a Fibonacci graceful labeling.
Proof Let the graph G =< C 3m : P 2 : C 3n > is obtained by joining a vertex of a cycle C n with a vertex of a cycle C m by an edge.
Let the vertices of C 3m and C 3n in order be v 0 , v 1 , v 2 , • • • , v 3m−1 and u 0 , u 1 , u 2 , • • • , u 3n−1 respectively.Let u o and v 0 be joined by an edge e.Then the vertex set of the graph is In view of the above defined labeling pattern f admits a Fibonacci graceful labeling for G.
That is, G is a Fibonacci graceful graph.Let the vertices of C 3m and C 3n be v In view of the above defined labeling pattern f admits a Fibonacci graceful labeling of the graph G.That is, G is a Fibonacci graceful graph.
Illustration 2.4 The Fibonacci graceful labeling of the graph joining a vertex of C 9 and a vertex of C 6 by a path P 3 is as shown in Fig. 3.
In view of the above defined labeling pattern f admits a Fibonacci graceful labeling of the graph G.That is, G is a Fibonacci graceful graph.
10846 4181 8362 5778 Fig. 4 Theorem 2.7 An arbitrary path union of k−copies of cycles C 3m is a Fibonacci graceful graph.
Proof Let the graph G be obtained by attaching cycles C i 3ni of length 3n i at each of the vertices v i of a path In view of the above defined labeling pattern f admits a Fibonacci graceful labeling for graph G.That is, G is a Fibonacci graceful graph.
In view of the above defined labeling pattern f admits a super Fibonacci graceful labeling of the graph G.That is, G is a super Fibonacci graceful graph. Thus Here each vertex label is either zero or a Fibonacci number at the most F q and each edge label is also a Fibonacci number at the most F q .In view of the above defined labeling pattern f admits a super Fibonacci graceful labeling for graph G.That is, G is a super Fibonacci graceful graph.
Case 3 If n ≡ 2 ( mod 3) then n = 3m + 2 for some positive integer m.Consider the graph G obtained from C 3m+2 by adding an edge v 0 v 3m−1 and one more edge v 3m v 3m+2 incident to the vertex v 3m and a new vertex v 3m+2 .Then the number of edges of G is In view of the above defined labeling pattern f admits a Fibonacci graceful labeling for graph G.That is, G is a Fibonacci graceful graph.Remark 3.4 In Case 3, if n ≡ 2( mod 3) then f (v 3m+2 ) = 2F 3m+2 which is not a Fibonacci number.Therefore such embedding is not a super Fibonacci graceful.Thus to embed a cycle C n with n ≡ 2( mod 3) as a subgraph of a super Fibonacci graceful graph remains an open problem.Illustration 3.5 A super Fibonacci graceful embedding of the cycle C 7 is shown in Fig. 7.
Fig. 7 Illustration 3.6 A Fibonacci graceful embedding of the cycle C 8 is shown in Fig. 8. Proof Let the graph G be obtained by taking one point union of k cycles C i 3ni of order 3n i for each In view of the above defined labeling pattern f admits a super Fibonacci graceful labeling of the graph G.That is, G is a super Fibonacci graceful graph.
Fig. 9 §4.Concluding Remarks Here we investigate four new results corresponding to Fibonacci graceful labeling and three new results corresponding to super Fibonacci graceful labeling of graphs.Analogous results can be derived for other graph families and in the context of different graph labeling problems.

Introduction and Definitions
We begin with simple, finite, undirected and non-trivial graph , with vertex set

 
V G and

 
E G are the cardinality of vertex set and edge set respectively.For various graph theoretic notations and terminology we follow Gross and Yellen [1] while for number theory we follow Niven and Zuckerman [2].We will give brief summary of definitions and other information which are useful for the present investigations.
Definition 1.1:If the vertices of the graph are assigned values subject to certain conditions then it is known as graph labeling.
Vast amount of literature is available in printed as well as in electronic form on different types of graph labeling.More than 1300 research papers have been published so far in last four decades.For a dynamic survey of graph labeling problems along with extensive bibliography we refer to Gallian [3].

Definition 1.2: A prime labeling of a graph is an injective function
such that for every pair of adjacent vertices u and , .The graph which admits a prime labeling is called a prime graph.
The notion of a prime labeling was originated by Entringer and was discussed in a paper by Tout et al. [4].Many researchers have studied prime graphs.For e.g.Fu and Huang [5]   , For the vertex we have the following possibilities: If the proof is similar as discussed above.
i for some .Then in i v , is adjacent to all the vertices except and Then clearly f is an injection.For an arbitrary edge of we claim that .
as and are consecutive positive integers; Thus in each of the possibilities the graph v under consideration admits a prime labeling.i.e.
is a prime graph.
Case 2: Let 0 be the apex vertex and 1 2 be the consecutive pendant vertices of be the graph obtained by switching the pendant vertex every vertex i other than 0 and n is adjacent to 0 and n v only.By Bertrand's postulate of number theory there exists at least one prime In view of the pattern defined above f admits a prime labeling on .Hence is a prime graph.

K
is the disjoint union of and   Then clearly f is an injection.
For an arbitrary edge of we claim that .be consecutive rim vertices of n and 0 be the apex vertex of .Let G be the graph obtained by switching the vertex .
, , , For an arbitrary edge of we claim that .
 is a positive integer less than the prime number . 1 n  Thus in each of the possibilities the graph under consideration admits a prime labeling.i.e. is a prime graph.,v 3, ,8 be a prime labeling.As 7 is adjacent to four vertices and 0 is adjacent to six vertices 1 2 3 4 5 6 , the labels of 7 v and 0 can not be even.Moreover we have to distribute four even labels among six vertices.Therefore at least two adjacent vertices from 1 2 3 4 5 6 will receive the even labels which contradicts the fact that f is a prime labeling.We noticed that it is not easy to discuss the prime labeling of a graph obtained by switching any rim vertex of n when W 1 n  is a composite number.However we prove a following result and pose a conjecture.
Theorem 2.14: Switching of a rim vertex in n is a not a prime graph if W 1 n  is an even integer greater than 9.
Proof: Let 1 2 be consecutive rim vertices and 0 be the apex vertex of n .Let be the graph obtained by switching the vertex .If possible assume that there exists a prime labeling the number of even integers is equal to the number of odd integers in

If a vertex
is adjacent to at least vertices then it cannot be labeled with even integer otherwise each of its with odd integers.Consequently there should be at least odd integers in {1 and at the most even integers.However, there are at least 5 even integers in , a contradiction.
Consequently each of the vertices 0 and n v have at least neighbours must be labeled with an odd integer.Therefore the remaining vertices 1 2 1 , , v v , n v   forms a path in and these vertices will receive 3 odd labels and 1 2 n  even labels.If L and M denote the number of vertices with even labels and number of vertices with odd labels respectively in then . Hence there must be two adjacent vertices in 1 n which will receive even labels which contradicts our assumption that P  f is a prime labeling of .G Conjecture 2.15: The graph obtained by switching of a rim vertex in n is a not a prime graph if W 1 n  is a composite odd integer greater than 9.

Concluding Remarks
The study of prime numbers is of great importance as prime numbers are scattered and there are arbitrarily large gaps in the sequence of prime numbers.If these characteristics are studied in the frame work of graph theory then it is more challenging and exciting as well.
Here we investigate several results on prime labeling of graphs and pose a conjecture.This discussion becomes more relevant as it is carried out in the context of a graph operation namely switching of a vertex.We have studied switching invariant behaviour of some standard graphs.

Introduction
We begin with finite, undirected and non-trivial graph with the vertex set and the edge set .Throughout this work n denotes the cycle with vertices and denotes the path on vertices.In the wheel For various graph theoretic notation and terminology we follow Gross and Yellen [1] while for number theory we follow Burton [2].We will give brief summary of definitions and other information which are useful for the present investigations.
Definition 1.1:If the vertices are assigned values subject to certain condition(s) then it is known as graph labeling.
Vast amount of literature is available in printed as well as in electronic form on different kind of graph labeling problems.For a dynamic survey of graph labeling problems along with extensive bibliography we refer to Gallian [3].
Definition 1.2: A prime labeling of a graph is an injective function such that for every pair of adjacent vertices and , .The graph which admits a prime labeling is called a prime graph.
The notion of a prime labeling was originated by Entringer and was discussed in a paper by Tout et al. [4].Many researchers have studied prime graphs.For e.g.Fu and Huang [5]  K two vertices can be identified in following two possible ways: Case 1: The apex vertex 0 is identified with any of the pendant vertices (say ).Let the new vertex be and the resultant graph be .
and .Define as Obviously f is an injection and for every pair of adjacent vertices and v of .Hence is a prime graph., , , : f is an injection.For an arbitrary edge of 1 G we claim that e uv    ( ), ( ) as i and gcd j v are adjacent vertices in the prime graph with the prime labelling G f .Thus in all the possibilities 1 f admits a prime labeling for .Hence is a prime graph.
is a prime labeling for with .Thus For let be the apex vertex and be 1, 2, , 1 Case 2: If x is one of the pend ant vertices then  , w e and the remaining 1 n        tively.For various graph theoretic notation and terminology we follow West [14] and for number theory we follow Burton [1].We give brief summary of definitions and other information which are useful for the present investigation.The graph which admits a prime labeling is called a prime graph.
The notion of a prime labeling was originated by Entringer and it was discussed by Tout et al [6].
Fu and Huang [3] proved that P n and K 1,n are prime graphs.Lee et al [5] proved that W n is a prime graph if and only if n is even.Deretsky et al [2] proved that C n is a prime graph.Vaidya and Kanani [9] discussed prime labeling of some cycle related graphs.A stronger concept of k-prime labeling is also introduced by Vaidya and Prajapati [10].The switching invariance of various graphs is discussed by Vaidya and Prajapati [11] and the same authors introduced the concept of strongly prime graph in [12].
A variant of prime labeling known as vertex-edge prime labeling is also introduced by Venkatachalam and Antoni Raj in [13].

Figure 2 .
Figure 2. The disjoint union of W 8 and P 6 and its prime labeling

Figure 3 .
Figure 3. Graph obtained by identifying a rim vertex of W 8 with an end vertex of P 6 and its prime labeling.

Figure 4 .
Figure 4.The graph obtained by identifying the apex vertex of W 8 with end vertex of P 6 and its prime labeling

Figure 5 .
Figure 5. Graph obtained by identifying the apex vertices of f 3 , f 4 , f 5 and its prime labeling.

Figure 6 .
Figure 6.The graph obtained by identifying the apex vertices of W 6 and W 8 and its prime labeling.

Figure 7 .
Figure 7.A 5-ply and its prime labeling

Illustration 1 . 4
The Fibonacci graceful labeling of K 1,6 and C 6 are shown in Fig.1

FibonacciDefinition 1 . 6 Definition 1 . 7
and Super Fibonacci Graceful Labelings of Some Cycle Related Graphs 61 graph there are two vertices having labels 0 and F q and these vertices are adjacent.The graph obtained by identifying a vertex of a cycle C n with a vertex of a cycle C m is the graph with |V | = m + n − 1, |E| = m + n and is denoted by < C n : C m >.The graph G =< C n : P k : C m > is the graph obtained by identifying one end vertex of P k with a vertex of C n and the other end vertex of P k with a vertex of C m .

Illustration 2 . 2
The Fibonacci graceful labeling of the graph joining a vertex of C 9 and a vertex of C 6 by an edge is as shown in Fig.2.

Fibonacci
and Super Fibonacci Graceful Labelings of Some Cycle Related Graphs

Illustration 3 . 2
The super Fibonacci graceful labeling of < C 9 : C 6 > is as shown in Fig.6.

Illustration 3 . 8 AF 10 F 10 F 11 F 12
super Fibonacci graceful labeling of the one point union of three cycles C 3 , C 6 and C 3 is as shown in Fig.9.0 claim the following cases are to be considered.a) If for some obtained by switching of any vertex in are prime graphs.Hence the result.

5 :
The prime labeling of the graph obtained by switching a pendant vertex of is shown in the Figure1.

5 P 2 . 6 :
IllustrationThe prime labeling of the graph obtained by switching a vertex of 5 which is not a pendant vertex is shown in the Figure2.

K 2 . 8 : 6 K is shown in the Figure 3 . Theorem 2 . 9 :Figure 1 .Figure 2 .Figure 3 .
Figure 1.The prime labeling of the graph obtained by switching a pendant vertex of 5 P .

Illustration 2 . 10 : 6 W 2 . 11 :
Thus in each of the possibilities the graph admits a prime labeling.i.e. is a prime graph.G G The prime labeling of the graph obtained by switching the apex vertex of is shown in the Figure 4.Theorem Switching of a rim vertex of is a prime graph if is a prime number.

Illustration 2 . 12 :
The prime labeling of the graph obtained by switching a rim vertex of is shown in the Figure5.

Figure 4 . 6 WFigure 5 . 6 W
Figure 4.The prime labeling of the graph obtained by switching the apex vertex of .6 W

n
 neighbours should receive the labels Copyright © 2012 SciRes.OJDM S. K. VAIDYA ET AL.Copyright © 2012 SciRes.OJDM 20 4 the apex vertex and the vertices corresponding to are called the rim vertices, where .The star 1,n n C 3 n  K is a graph with one vertex of degree called apex and vertices of degree one are called pendant vertices.Throughout this paper n n ( ) V G and ( ) E G denote the cardinality of vertex set and edge set respectively.

1 ( 2 :
f u  .Obviously f is an injection and   ), ( ) f u f v gcd  for every pair of adjacent vertices Copyright © 2012 SciRes.OJDM ISSN 2161-7635 S. K. VAIDYA, U. M. PRAJAPATI 100 u and v of G .Hence is a prime graph.G Case Any two of the pendant vertices (say 1 n v  and n ) are identified.Let the new vertex be 1 n and the resultant graph be G.So in G, , for

2 : 2 . 3 :u
The prime labeling of the graph obtained by identifying the apex vertex with a pendant vertex of 1,7 K is shown in Figure 1.Illustration The prime labeling of the graph obtained by identifying two of the pendant vertices of 1,7 K is shown in Figure 2. Theorem 2.4: If is a prime and is a prime graph of order then the graph obtained by identifying two vertices with label 1 and is also a prime graph.p G p p Proof: Let f be a prime labeling of and i be the label of the vertex i v for be the new vertex of the graph 1 which is obtained by identifying gcd f u f v  .To prove our claim the following cases are to be considered.

1 1 Illustration 2 . 5 : 6 :Figure 1 .
Figure 1.The prime labeling of the graph obtained by identifying the apex vertex with a pendant vertex in K 1,7 .

Figure 2 .
Figure 2. The prime labeling of the graph obtained by identifying two of the pendant vertices in K 1,7 .

Figure 3 .
Figure 3.The prime labeling of a graph of order five.

Figure 4 .Illustration 2 . 7 : 3 .
Figure 4.The prime labeling of the graph obtained by identifying the vertices of Figure 3 with label 1 and 5.

Figure 5 .
Figure 5.A prime labeling of P 5 .

Figure 6 .
Figure 6.A prime labeling of the graph obtained by identifying v 1 and v 2 of P 5 of Figure 5.

Figure 7 .
Figure 7.A prime labeling of the graph obtained by identifying v 1 and v 3 of P 5 of Figure 5.

Figure 8 .
Figure 8.A prime labeling of the graph obtained by identifying v 1 and v 4 of P 5 of Figure 5.

6 : 3 . 7 :
It is possible to assign label 1 to arbitra rongly prime graph.Thus from the cases d is n me graph.Illustration ry vertex of 5 P in order to obtain different prime labeling as shown Figures 13-18.Theorem Every cycle is a st in

fFigure 17 . A prime labeling of P 5 having as label 1. v 5 Figure 18 .
Figure 17.A prime labeling of P 5 having as label 1. v 5

v 5 Figure 10 .
A strongly prime labeling of K 3 .

Figure 11 .
Figure 11.A prime labeling of K e 4  .

Figure 13 . A prime labeling of P 5 Figure 14 . A prime labeling of P 5
Figure 12.is not a strongly prime graph.K e 4 

Figure 1 (
Figure 15.A prime labeling of P 5 having as label 1. v 3

Figure 16 .Figure 19 . A prime labeling of K 1
Figure 16.A prime labeling of P 5 having v 4 as label 1.

Figure 20 .where p is the largest prime less than or equal 1 . 1 (
Figure 20.A prime labeling of K 1,7 with a pendant vertex as

Figure 21 .
Figure 21.A prime labeling of W 8 with the ap vertex as ex label 1.

Figure 22 . 1 Introduction
Figure 22.A prime labeling of W 8 with a rim vertex as label 1.

Definition 1 . 1 .Definition 1 . 2 .
If the vertices of the graph are assigned values subject to certain condition(s) then it is known as graph labeling.A prime labeling of a graph G is an injective function f : V (G) → {1, 2, . . ., |V (G)|} such that for every pair of adjacent vertices u and v, gcd(f (u), f (v)) = 1.
The graph obtained by joining a vertex of C 3m and a vertex of C 3n by a path P k admits Fibonacci graceful labeling.
Proof Let the graph G =< C 3m : P k : C 3n > is obtained by joining one vertex of a cycle C n with one vertex of a cycle C m by a path of length k.Let the vertices of C 3m and C 3n Illustration 2.6 A Fibonacci graceful labeling of the graph obtained by joining a vertex of C 9 and a vertex of C 6 by a path P 6 is shown in the following Fig.4.
have proved that n and P is a prime graph.

. Prime Labeling of Some Graphs
have proved that n and P