Prime Labeling for Some Cycle Related Graphs

Here we investigate prime labeling for some cycle related graphs. We also discuss prime labeling in the context of some graph operations namely fusion, duplication and vertex switching in cycle Cn.


Introduction
We begin with finite, connected and undirected graph G = (V(G), E(G)) without loops and multiple edges.Here elements of sets V(G) and E(G) are known as vertices and edges respectively.In the present work C n denotes cycle with n vertices and N(v) denotes the set of all neighboring vertices of v.For all other terminology and notations in graph theory we follow (Gross, J., 1999) where as for number theory we follow (Niven, I., 1972).We will give brief summary of definitions which are useful for the present investigations.
Definition 1.1 If the vertices of the graph are assigned values subject to certain conditions is known as graph labeling.Enough literature is available in printed as well as in electronic form on different types of graph labeling and more than 1000 research papers have been published so far in past four decades.A current survey of various graph labeling problems can be found in (Gallian, J., 2009).
Following three are the common features of any graph labeling problem.
(1) a set of numbers from which vertex labels are assigned; (2) a rule that assigns a value to each edge; (3) a condition that these values must satisfy.
The present work is targeted to discuss one such labeling known as prime labeling defined as follows.
The notion of a prime labeling was introduced by Roger Entringer and was discussed in a paper by (Tout, A,1982, p.365-368).Many researchers have studied the prime graphs.For e.g. in (Fu, H., 1994, p.181-186) have proved that path P n on n vertices is a prime graph.In (Deretsky, T., 1991, p.359-369) have proved that the cycle C n on n vertices is a prime graph.In (Lee, S., 1988, p.59-67) have proved that wheel W n is a prime graph if and only if n is even.Around 1980 Roger Etringer conjectured that All trees have prime labeling which is not settled till today.The prime labeling for planar grid is investigated by (Sundaram, M., 2006, p.205-209).
Definition 1.3An independent set of vertices in a graph G is a set of mutually non-adjacent vertices.
Definition 1.4 The independence number of a graph G is the maximum cardinality of an independent set of vertices.It is denoted by ind(G) or α(G).
Definition 1.5 Let u and v be two distinct vertices of a graph G.A new graph G 1 is constructed by identifying(fusing) two vertices u and v by a single vertex x is such that every edge which was incident with either u or v in G is now incident with x in G 1 .
In other words a vertex v k is said to be duplication of v k if all the vertices which are adjacent to v k are now adjacent to v k Definition 1.7 A vertex switching G v of a graph G is obtained by taking a vertex v of G, removing all the edges incident with v and adding edges joining v to every vertex which are not adjacent to v in G.
In the present work we prove that the graphs obtained by identifying any two vertices, duplicating arbitrary vertex and switching of any vertex in cycle C n admit prime labeling.We also prove that the graph obtained by path union of cycle C n is a prime graph except for odd n.In addition to this we show that the graph obtained by joining two copies of cycle C n by a path of arbitrary length is a prime graph except n and k both are odd.

Main Results
Theorem 2.1 The graph obtained by identifying any two vertices v i and v j (where d(v i , v j ) ≥ 3) of cycle C n is a prime graph.
Proof: Let C n be the cycle with vertices v 1 , v 2 , . . ., v n and the vertex v 1 be fused with v m where m ≤ n/2 .Denote the resultant graph as G.Here we note that |V(G)| = n − 1.It is obvious that identifying two vertices of cycle C n produces connected graph which includes two edge disjoint cycles C m−1 and C n−m+1 .
In each case described above the graph under consideration admits prime labeling i.e.G is a prime graph.

Remark 2.2 (i)when m > n
2 identifying two vertices will repeat all the graphs which are already considered earlier for m ≤ n 2 .(ii)when d(v i , v j ) < 3 then fusion yields a graph which is not simple and it is not desirable for prime labeling.Proof: Let v k be any arbitrary vertex of cycle C n , v k be its duplicated vertex and G be the graph resulted due to duplication of vertex v k .Then |V(G)| = n + 1. Define labeling f : V(G) → {1, 2, . . ., n + 1} as follows.
Thus function defined above provides prime labeling for graph G i.e. graph G under consideration is a prime graph.Define labeling f : V(G v ) → {1, 2, . . ., n} as follows.
Thus f admits a prime labeling and consequently G v is a prime graph.
Illustration 2.7 Consider a graph G obtained by switching the vertex v 1 of cycle C 9 .The corresponding prime labeling is shown in Fig 3.
Theorem 2.8 The graph obtained by the path union of finite number of copies of cycle C n is a prime graph except for odd n.
Proof: Let G be the path union of cycle C n and G 1 , G 2 , . . ., G k be k copies of the cycle C n .We note that |V(G)| = nk.Let us denote the successive vertices of the graph G i by u i1 , u i2 , . . ., u in .Let e i = u i1 u (i+1)1 be the edge joining G i and G i+1 for i = 1, 2, . . ., k − 1.
Thus we proved that the graph obtained by the path union of finite number of copies of cycle C n is a prime graph except for odd n.Illustration 2.9 Consider a graph G obtained by path union of three copies of the cycle C 10 .It is the case related to n ≡ 0(mod2) and k ≡ 1(mod2).The prime labeling is as shown in Fig 4 .Theorem 2.10 The graph obtained by joining two copies of cycle C n by a path P k is a prime graph except n and k both are odd.
Proof: Let G be the graph obtained by joining two copies of cycle C n by a path P k .We note that |V(G)| = 2n + k − 2. Let u 1 , u 2 , . . ., u n be the vertices of first copy of cycle C n and v 1 , v 2 , . . ., v n be the vertices of second copy of cycle C n .Let w 1 , w 2 , . . ., w k be the vertices of path P k with u 1 = w 1 and v 1 = w k .

Illustration 2. 3
Consider a graph G obtained by identifying the vertex v 1 with v 6 of cycle C 11 .It is the case related to n ≡ 1(mod2).The labeling is as shown in Fig 1.

Theorem 2. 4
The graph obtained by duplicating arbitrary vertex of cycle C n is a prime graph.

Illustration 2. 5
Consider a graph G obtained by duplicating a vertex v 1 in cycle C 6 .The labeling is as shown in Fig 2.

Theorem 2. 6
The switching of any vertex in cycle C n produces a prime graph.Proof: Let G = C n and v 1 , v 2 , . . ., v n be the successive vertices of C n and G v denotes the vertex switching of G with respect to the vertex v.It is obvious that |V(G v )| = 2n − 5. Without loss of generality we initiate the labeling from v 1 and proceed in the clockwise direction.

Figure 2 .
Figure 2. Vertex duplication in C 6 and Prime labeling