Some Results on Prime Labeling

In the present work we investigate some classes of graphs and disjoint union of some classes of graphs which admit prime labeling. We also investigate prime labeling of a graph obtained by identifying two vertices of two graphs. We also investigate prime labeling of a graph obtained by identifying two edges of two graphs. Prime labeling of a prism graph is also discussed. We show that a wheel graph of odd order is switching invariant. A necessary and sufficient condition for the complement of n W to be a prime graph is investigated.


Introduction
We begin with simple, finite, undirected and non-trivial graph ( ) 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 n C denotes the cycle with n vertices and n P denotes the path of n vertices.In the wheel K is called the apex vertex and the vertices corresponding to n C are called the rim vertices.For various graph theoretic notations and terminology we follow Gross and Yellen [1] whereas for number theory we follow D. M. 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 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 J. A. Gallian [3].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 of order n is an injective function such that for every pair of adjacent vertices u and v , ( , 1 gcd f u f v = .The graph which admits prime labeling is called a prime graph.
The notion of prime labeling was originated by Entringer and was discussed in A.Tout [4].Many researchers have studied prime graphs.It has been proved by H. L. Fu and C. K. Huang [5] that n P is a prime graph.It has been proved by S. M. Lee [6] that wheel graph n W is a prime graph if and only if n is even.T. Deretsky [7] has proved that cycle n C is a prime graph.Definition 1.3: A vertex switching v G of a graph G is the graph obtained by taking a vertex v of G , removing all the edges incident to v and adding edges joining to every other vertex which is not adjacent to v in G .
Definition 1.4: A prime graph is said to be switching invariant if for every vertex v of G , the graph v G obtained by switching the vertex v in G is also a prime graph.
Definition 1.5:For two graphs ( ) ( ) , u v and ( )  G is a prime graph with order n , where n is even and 2 G is a graph with order 3 then disjoint union of 1 G and 2 G is a prime graph.Proof:  ( )


Clearly h is an injective function.To prove h is a prime labeling of G we have the following cases: Here d is an odd natural number as n is even and at least one of ( ) i g v and ( ) ) ( ) ( ) ( ) ) Thus G admits prime labeling.So G is a prime graph.S. K. Vaidya and U. M. Prajapati [8] has proved that if 1 4 n ≥ is an even integer and 2 n is a natural num- ber, then the graph obtained by identifying any of the rim vertices of a wheel 1 n W with an end vertex of a path graph 2 n P is a prime graph.But in the following theorem we have prove that if 1 n is odd then also it is prime.Theorem 2.3: n n p + = , where p is prime then the graph obtained by identifying one of the rim vertices of Clearly f is an injective function.Let e be an arbitrary edge of G .To prove f is a prime labeling of G we have the following cases: Case 1: Case 5: Thus G admits a prime labeling.So G is a prime graph.Theorem 2.4: A path 1 m P + and m copies of cycle n C are given, then the graph obtained by identifying each edge of m P with an edge of a corresponding copy of the cycle n C is prime.Proof: Let G be a graph obtained by identifying an edge 1, , Let e be an arbitrary edge of G .To prove f is a prime labeling of G we have the following cases: Case 1: Thus G is a prime graph.So G is a prime graph.Theorem 2.5: A cycle m C and m copies of a cycle n C are given, then the graph obtained by identifying each edge of m C with an edge of corresponding copy of the cycle n C is prime.C with an edge , .
Let e be an arbitrary edge of G .To prove f is a prime labeling of G we have the following cases: Case 1: Case 3: Case 5:  .Thus G admits a prime labeling.So G is a prime graph.S. K. Vaidya and U. M. Prajapati [9] have proved that switching the apex vertex in n W is a prime graph and switching a rim vertex in n W is a prime graph if 1 n + is prime.But in the following theorem we have proved that n W is switching invariant if n is even.

Theorem 2.6: 2n
W is switching invariant.Proof: be rim vertices and 0 v be an apex vertex of 2n W .According to the degree of vertices of 2n W we can take the following cases.Case 1: Let G be a graph obtained by switching a rim vertex 2n v .Let p be a largest prime less than 2n .Define a function ( ) for 2 .
Clearly f is an injective function.Let e be an arbitrary edge of G .To prove f is a prime labeling of G we have the following cases: i n gcd f v f v gcd p i p gcd i p = + = = , as p is the largest prime less than n 2 .


Clearly f is an injective function.It can be easily verified that f is a prime labeling.Thus from both the cases it follows that G is a prime graph.Theorem 2.7: The complement of n W is prime if and only if 3 6.n ≤ ≤ Proof: We can easily see that n W is prime for 3, 4, 5 n = and 6 from Figure 1.

Now if 7
n ≥ then ( ) 3 4 n − ≥ and every rim vertex of n W is adjacent to other ( )  even numbers to assign 1 n + vertices.If one of the rim vertices is labeled as even number then other 3 − n vertices can not be labeled as even number.Also remaining two rim vertices are adjacent, so only one of them can be labeled as even number.The apex vertex can also be labeled as even number.Thus maximum three vertices can be labeled as even number.But if 7 n ≥ then we have 4 or more even numbers to label.So it is not possible.
 Clearly f is an injective function.Let e be an arbitrary edge of G .To prove f is a prime labeling of G we have the following cases: Case 1: ( ) 1, 1 as 1 so 1 and is a prime.Thus G admits a prime labeling.So G is a prime graph.Theorem 2.9: If 3 n ≥ is an odd integer then the prism graph Clearly f is an injective function.Let e be an arbitrary edge of G .To prove f is a prime labeling of G we have the following cases: Case 4: If Thus G admits a prime labeling.So G is a prime graph.Theorem 2.11: A graph obtained by joining every rim vertex of a wheel graph

1 nW
loss of generality we can assume that ( ) , G V E be the graph obtained by identifying a rim vertex 1 u of with an end vertex 1 v of 2 n P .Define


Clearly f is an injective function.Let e be an arbitrary edge of G .To prove f is a prime labeling of G we have the following cases:

C
− is a prime graph, where p is a prime number not less than 3.

C
− .Let G be the graph obtained by joining a vertex i v of