Some New Families of Prime Cordial Graphs

In this paper some new families of prime cordial graphs are investigated. We prove that the square graph of path Pn is a prime cordial graph for n = 6 and n ≥ 8 while the square graph of cycle Cn is a prime cordial graph for n ≥ 10. We also show that the shadow graph of K1,n for n ≥ 4 and the shadow graph of Bn,n are prime cordial graphs. Moreover we prove that the graphs obtained by mutual duplication of a pair of edges as well as mutual duplication of a pair of vertices from each of two copies of cycle Cn admit prime cordial labeling.


Introduction
We begin with simple, finite, connected and undirected graph G = (V(G), E(G)) with p vertices and q edges.For standard terminology and notations we follow (Gross, J., Yellen, J. , 1999).We will provide brief summary of definitions and other information which are necessary for the present investigations.
1.1 Definition If the vertices are assigned values subject to certain condition(s) then it is known as graph labeling.
For a dynamic survey on various graph labeling problems along with extensive bibliography we refer to (Gallian J. A., 2010).A graph G is cordial if it admits cordial labeling.

Definition
The concept of cordial labeling was introduced in (Cahit, I., 1987, p.201-207).After this many researchers have investigated graph families or graphs which admit cordial labeling.Some labeling schemes are also introduced with minor variations in cordial theme.Product cordial labeling, total product cordial labeling and prime cordial labeling are among mention a few.The present work is focused on prime cordial labeling.The concept of prime cordial labeling was introduced in (Sundaram, M., Ponraj, R. and Somasundram, S., 2005, p.373-390) and they have investigated several results on prime cordial labeling.Prime cordial labeling in the context of graph operations is discussed in (Vaidya, S. K. and Vihol, P. L., 2010, p.119-126).The same authors have discussed prime cordial labeling for some cycle related graphs in (Vaidya, S.K. and P.L.Vihol, 2010, p. 223-232).In the present work we will investigate some new families of prime cordial graphs.

Definition
1.6 Definition For a simple connected graph G the square of graph G is denoted by G 2 and defined as the graph with the same vertex set as of G and two vertices are adjacent in G 2 if they are at a distance 1 or 2 apart in G.

Definition
The shadow graph D 2 (G) of a connected graph G is obtained by taking two copies of G as G and G .Join each vertex u in G to the neighbours of the corresponding vertex u in G .
Case 1: n = 6, 8, 9 The graphs P 2 6 , P 2 8 and P 2 9 are to be dealt separately and their prime cordial labeling are shown in Figure 1.Case 2: n is even, n ≥ 10 Thus in all the above cases we have |e f (0) − e f (1)| ≤ 1.
Hence P 2 n is a prime cordial graph for n = 6 and n ≥ 8.
Then obviously e f (0) = 7.In order to satisfy edge condition for prime cordial graph it is essential to label eight edges with 0 and eight edges with 1 out of sixteen edges.But in all the possible assignments we are getting at most seven edge labels with 0 and at least nine edge labels with 1.So |e f (0) − e f (1)| > 1. Therefore C 2 8 is not a prime cordial graph.Now for the graph C 2 9 in order to satisfy edge condition for prime cordial graph it is essential to label nine edges with 0 and nine edges with 1 out of eighteen edges.But in all the possible assignments we are getting at most seven edge labels with 0 and at least eleven edge labels with 1.So |e f (0) − e f (1)| > 1. Therefore C 2 9 is not a prime cordial graph.Hence C 2 n is not a prime cordial graph for n ≤ 9.

Theorem: C 2
n is a prime cordial graph for n ≥ 10.
, 2, 3, . . ., n}, we consider following three cases.Case 1: n = 10 The graph C 2 10 is to be dealt separately and its prime cordial labeling is shown in Figure 3. Case 2: n is odd, n ≥ 11 Sub Case 1: n ≡ 1(mod 3) or n ≡ 2(mod 3) In view of the labeling pattern defined above, we have e f (0) = e f (1) = n.
Sub Case 2: n ≡ 0(mod 3) Proof: Consider two copies of K 1,n .Let v 1 , v 2 , . . ., v n be the pendant vertices of the first copy of K 1,n and v 1 , v 2 , . . ., v n be the pendant vertices of second copy of K 1,n with v and v are the respective apex vertices.Let G be the graph In graph D 2 (K 1,2 ), in order to satisfy edge condition for prime cordial labeling it is essential to label four edges with 0 and four edges with 1.But all the possible assignment of vertex labels give rise to 0 labels for at most three edges and 1 labels for at least five edges.Thus |e f (0) − e f (1)| = 2 > 1. Hence D 2 (K 1,2 ) is not a prime cordial graph.
To satisfy edge condition for prime cordial labeling in the graph D 2 (K 1,3 ) it is essential to label six edges with 0 and six edges with 1.But all the possible assignments of vertex labels give rise to 0 labels for at most five edges and 1 labels for at least seven edges.Thus |e f (0) − e f (1)| = 2 ≥ 1. Hence D 2 (K 1,3 ) is not a prime cordial graph.
Hence D 2 (K 1,n ) is a not a prime cordial graph for n = 2, 3.
2.9 Illustration: Prime cordial labeling of the graph D 2 (K 1,5 ) is shown in Figure 5.

Theorem: D
The prime cordial labeling of B 2,2 is as shown in the Figure 6.
odd numbers between 5 to 4n + 3 except p and 3p.In view of the labeling pattern defined above we have e f (0) = e f (1) = 2(2n + 1).In order to satisfy the edge condition for the graph obtained by mutual duplication of a pair of edges in C 3 it is essential to label five edges with 0 and five edges with 1 out of ten edges.But all the possible assignment of vertex labels give rise to label 0 for at most four edges and label 1 for at least six edges.Thus |e f (0) − e f (1)| = 2 > 1. Hence the graph obtained by mutual duplication of a pair of edges in C 3 is not a prime cordial graph.

Thus in both the cases we have |e
In order to satisfy the edge condition for the graph obtained by mutual duplication of a pair of edges in C 4 it is essential to label six edges with 0 and six edges with 1 out of twelve edges.But all the possible assignment of vertex labels give rise to label 0 for at most five edges and label 1 for at least seven edges.Thus |e f (0) − e f (1)| = 2 > 1. Hence the graph obtained by mutual duplication of a pair of edges in C 4 is not a prime cordial graph.
Hence the graph obtained by mutual duplication of a pair of edges in C n is not a prime cordial for n = 3, 4.

Theorem:
The graph obtained by mutual duplication of a pair of edges in C n admits prime cordial labeling for n ≥ 5.
Proof: Let v 1 , v 2 , . . ., v n be the vertices of first copy of cycle C n and u 1 , u 2 , . . ., u n be the vertices of the second copy of cycle C n .Let G be the graph obtained by mutual duplication of a pair of edges each respectively form each copy of cycle C n .Then |V(G)| = 2n and |E(G)| = 2n + 4. To define f : V(G) → {1, 2, 3, . . ., 2n + 4}, we consider following two cases.
Case 1: n is odd, n ≥ 5 Without loss of generality we may assume that the edge e = v n+1 2 v n+3 2 from the first copy of cycle C n and the edge e = u 1 u 2 from the second copy of cycle C n are mutually duplicated.
Without loss of generality we may assume that the edge e = v n 2 +1 v n 2 +2 from the first copy of cycle C n and the edge e = u 1 u 2 from the second copy of cycle C n are mutually duplicated.
In view of the labeling pattern defined above we have e f (0) = e f (1) = 2n + 2.

Thus in both the cases we have |e
Hence the graph obtained by mutual duplication of a pair of edges in C n admits prime cordial labeling for n ≥ 5.In order to satisfy the edge condition for the graph obtained by mutual duplication of a pair of vertices in C 3 it is essential to label five edges with 0 and five edges with 1 out of ten edges.But all the possible assignment of vertex labels give rise to label 0 for at most four edges and label 1 for at least six edges.Thus |e f (0) − e f (1)| = 2 > 1. Hence the graph obtained by mutual duplication of a pair of vertices in C 3 is not a prime cordial graph.

Illustration
In order to satisfy the edge condition for the graph obtained by mutual duplication of a pair of vertices in C 4 it is essential to label six edges with 0 and six edges with 1 out of twelve edges.But all the possible assignment of vertex labels give rise to label 0 for at most five edges and label 1 for at least seven edges.Thus |e f (0) − e f (1)| = 2 > 1. Hence the graph obtained by mutual duplication of a pair of vertices in C 4 is not a prime cordial graph.
Hence the graph obtained by mutual duplication of a pair of vertices in C n is not prime cordial for n = 3, 4.

Theorem:
The graph obtained by mutual duplication of a pair of vertices in C n admits prime cordial labeling for n ≥ 5.
Proof: Let v 1 , v 2 , . . ., v n be the vertices of the first copy of cycle C n and u 1 , u 2 , . . ., u n be the vertices of the second copy of cycle C n .Let G be the graph obtained by mutual duplication of a pair of vertices each respectively from each of two copies of C n .Then |V(G)| = 2n and |E(G)| = 2n + 4. To define f : V(G) → {1, 2, 3, . . ., 2n + 4} we consider following two cases.
Case 1: n is odd, n ≥ 5 Without loss of generality we may assume that the vertex v n+3 2 from the first copy of cycle C n and the vertex u 1 from the second copy of cycle C n are mutually duplicated.
Without loss of generality we may assume that the vertex v n+2 2 from the first copy of cycle C n and the vertex u 1 from the second copy of cycle C n are mutually duplicated.
In view of the labeling pattern defined above we have e f (0) = e f (1) = 2n + 2.
Hence the graph obtained by mutual duplication of a pair of vertices in C n admits prime cordial labeling for n ≥ 5.
2.17 Illustration: Prime cordial labeling of the graph obtained by mutual duplication of a pair of vertices in C 10 is shown in Figure 9.

Concluding Remarks
As every graph is not a prime cordial graph it is very interesting to investigate graph or graph families which admit prime cordial labeling.In this paper we have investigated some new results on prime cordial labeling.
A mapping f : V(G) → {0, 1} is called binary vertex labeling of G and f (v) is called the label of the vertex v of G under f .1.3Notations If for an edge e = uv, the induced edge labeling f* : E(G) → {0, 1} is given by f * (e) = | f (u) − f (v)|.Then v f (i) = number of vertices of G having label i under f e f (i) =number of edges of G having label i under f * where i = 0 or 1 1.4 Definition A binary vertex labeling f of a graph G is called a cordial labeling if |v f (0)−v f (1)| ≤ 1 and |e f (0)−e f (1)| ≤ 1.
Consider two copies of cycle C n .Then the mutual duplication of a pair of vertices v k and v k from each of two copies of cycle C n produces a new graph G such that N(v k )=N(v k ).1.10 Definition Consider two copies of cycle C n and let e k = v k v k+1 be an edge in the first copy of C n with e k−1 = v k−1 v k and e k+1 = v k+1 v k+2 be its incident edges.Similarly let e k = u k u k+1 be an edge in the second copy of C n with e k−1 = u k−1 u k and e k+1 = u k+1 u k+2 be its incident edges.The mutual duplication of a pair of edges e k and e k from each of two copies of cycle C n produces a new graph G in such a way that N 5In view of the labeling pattern defined above, we have e f (0) = e f (1) = n.Thus in the above cases, we have |ef (0) − e f (1)| ≤ 1.Hence C 2 n is a prime cordial graph for n ≥ 10.2.6 Illustration: Prime cordial labeling of the graph C 215 is shown in Figure4.2.7 Theorem: D 2 (K 1,n ) is a not a prime cordial graph for n = 2, 3.
2.11 Illustration: Prime cordial labeling of the graph D 2 (B 5,5 ) is shown in Figure7.2.12 Theorem: The graph obtained by mutual duplication of a pair of edges in C n is not a prime cordial graph for n = 3, 4.Proof: Let v 1 , v 2 , . . ., vn be the vertices of the first copy of cycle C n and u 1 , u 2 , . . ., u n be the vertices of the second copy of cycle C n .Denote by G the graph obtained from mutual duplication of a pair of edges respectively from each of two copies of cycle C n .Then |V(G)| = 2n and |E(G)| = 2n + 4.
: Prime cordial labeling of the graph obtained by mutual duplication of a pair of edges in C 9 is shown in Figure 8. 2.15 Theorem: The graph obtained by mutual duplication of a pair of vertices in C n is not a prime cordial graph for n = 3, 4. Proof: Let v 1 , v 2 , . . ., v n be the vertices of the first copy of cycle C n and let u 1 , u 2 , . . ., u n be the vertices of the second copy of cycle C n .Let G be the graph obtained by mutual duplication of a pair of vertices each respectively from each copy of cycle C n .Then |V(G)| = 2n and |E(G)| = 2n + 4.

FigureFigure 8 .Figure 9 .
Figure 2. P 2 12 and its prime cordial labeling If v 1 ,v 2 ,. . ., v n are the vertices of path P n then P 2 2 n is not a prime cordial graph for n = 3, 4, 5, 7. Proof: To investigate similar results for other graph families and in the context of different labeling problems is an open area of research.