On Signed Product Cordial Labeling

A new concept of labeling called the signed product cordial labeling is introduced and investigated for path graph, cycle graphs, star-K1,n, Bistar-Bn,n, and Some general results on signed product cordial labeling are studied. , n P n   3 , 3. n C n  


Introduction
If the vertices of the graph are assigned values subject to certain conditions then it is known as graph labeling.Most of the graph labeling problems have the following three common characteristics: a set of numbers for assignment of vertex labels, a rule that assigns a label to each edge and some condition(s) that these labels must satisfy.
Cordial labelings were introduced by Cahit [1] who called a graph G cordial if there is a vertex labeling such that the induced labeling , defined by , for all edges ( ) xy E G  and with the following inequalities holding: , where (respectively ) is the number of vertices (respectively, edges) labeled with .Sundaram and Somasundaram [2] introduced the notion of product cordial labelings.A product cordial labeling of a graph G with vertex set is a function ( ) the number of vertices labeled with 0 and the number of vertices labeled with 1 differ by at most 1, and the number of edges labeled with 0 and the number of edges labeled with 1 differ by at most 1.A graph with a product cordial labeling is called a product cordial graph.
For detail survey on graph labeling one can refer Gallian [3].Let be a graph.As given in [4] a mapping is called binary vertex labeling of G and ( , ) is called the label of the vertex v of G under f.For an edge , the induced edge labeling is given by Let be the number of vertices of G having labels 0 and 1 respectively under f and let be the number of edges having labels 0 and 1 respectively under In Section 2, we introduce the definition of signed product cordial labeling and work for few fundamental graphs.In Section 3 we prove that and bistar graph , n n are signed product cordial.Finally in Section 4, we prove the existence of signed product cordial labeling for some general graphs. ,

Signed Product Cordial Labeling
We now introduce the definition of signed product cordial labeling.Definition 2.1: A vertex labeling of graph G with induced edge labeling defined by ) 1  , where ( 1) f v is the number of vertices labeled with 1, ( 1) f e   is the number of edges labeled with -1 and is the number of edges labeled with 1.A graph G is signed product cordial if it admits signed product cordial labeling.
(1) e Theorem 2.2: The Path graph P n , n ≥ 2 admits signed product cordial labeling. Proof: be the vertex set and be the edge set of the path graph n .To define vertex labeling the following cases are to be considered.
The induced edge labeling f  is given by ( ) and ( 1 ( ) and ( gn ) have same sign with respect to the above labeling pattern we give the proof as follows, i) When 0 mod 4 n  The total number of vertices labeled with -1's are given by ( 1) 2 and the total number of vertices labeled with 1's are given by (1) 2 f .Therefore the total difference between the vertices labeled with -1's and 1's is . The total number of edges labeled with -1's are given by 2 1 e n    f and the total number of edges labeled with 1's are given by  * (1) 2 e n  f .Therefore the total difference between the edges labeled with -1's and 1's is * * ( 1) (1) 1 1    , differ by one.
( 1) (1) 2 ( 1) (1) 1 2 The total number of vertices labeled with -1's are given by   ( 1) 1 2 and the total number of vertices labeled with 1's are given by   (1) 1 2 . Therefore the total difference between the vertices labeled with -1's and 1's is The total number of edges labeled with -1's are given by and the total number of edges labeled with 1's are given by   . Therefore the total difference between the edges labeled with -1's and  , differ by zero.
Thus in each cases we have ( 1) (1) 1 be the edge set of the cycle graph C n .To define vertex labeling the following case is to be considered. ;1 The edge labeling is given by ( ) and ( ) have same sign 1 ( ) and ( ) have different sign In view of the above labeling pattern we have, Table 1.

Table 1. Vertex and edge conditions of a cycle graph.
n ( 1)

 
Proof: The star graph K 1,n is a tree obtained by adding n pendent edge to the center vertex.Let be the vertex set and the edge set is given by .To define vertex labeling for is given by When n is even and odd, the edge labeling is given by respectively.
In view of the above labeling pattern we have, Table 2.
Hence the star graph 1,n K admits signed product cordial labeling.

Signed Product Cordial Labeling for Special Graphs
In this section we prove the signed product cordial labeling for the graphs and the bistar graph ., and 1 2 3 be the vertex sets of the path graph and the edge set is given by , , The graph n has 2n vertices and edges.To define vertex labeling the following  ( 1) ( 1) (1) cases are to be considered.Case 1: when n is even for 1 i n   1 ; 1, 2 mod 4 ( ) 1; 0, 3 mod 4 The edge labeling are defined as follows for 1 1 if ( ) and ( ) have same sign 1 if ( ) and ( ) have different sign 1 if ( ) and ( ) have same sign 1 if ( ) and ( ) have different sign The vertex labeling is given by for 1 The edge labeling are defined as follows for 1 1 if ( ) and ( ) have same sign 1 if ( ) and ( ) have different sign 1 if ( ) and ( ) have same sign 1 if ( ) and ( ) have different sign In view of the above labeling pattern we have, Table 3.
Hence the graph admits signed product cordial labeling.
( 1) (1) Proof: Let 1 2 3 and 1 2 3 be the vertex sets of the graph n with 2n vertices and 2n edges.The edge set is given by , , The vertex labeling is defined by The edge labeling is given by for 1 if ( ) and ( ) have same sign 1 if ( ) and ( ) have different sign 1 if ( ) and ( ) have same sign ( ) 1 if ( ) and ( ) have different sign ( ) and ( ) have same sign ( ) 1 ( ) and ( ) have different sign In view of the above labeling pattern we have, the total number of vertices labeled with -1's are given by and the total number of vertices labeled with 1's are given by .Therefore the total difference between the vertices labeled with -1's and 1's is . The total number of edges labeled with -1's are given by f and the total number of edges labeled with 1's are given by (1) e   .Therefore the total difference between the edges labeled with -1's and 1's is ( 1) (1) 0    , differ by zero.
( 1) (1) ( 1) (1) Hence the cycle graph admits signed product cordial labeling., , , be the vertices of the bistar graph.The edge set is defined as We now define vertex labeling as The edge labeling is given by The total number of vertices labeled with -1's are given by ( 1) 1 f v n    and the total number of vertices labeled with 1's are given by .Therefore the total difference between the vertices labeled with -1's and 1's is . The total number of edges labeled with -1's are given by f and the total number of edges labeled with 1's are given by Therefore the total difference between the edges labeled with -1's and 1's is ( 1) (1) 1    , differ by one.In view of the above labeling pattern we have, ( 1) (1) 1 1 From the above labeling pattern we have    .Hence the bistar graph , admits signed product cordial labeling.
, , Example 3.4: Figure 1 illustrates the signed product cordial labeling for Bistar 5,5 .Among the eleven edges five edges receive the label +1 and six edges receive the label -1.

General Results on Signed Product Cordial Labeling
In this section we prove the Signed product cordial labeling for the some general graphs.Definition 4.1: The tree T@mK 1 is obtained by attaching m copies of K 1 to any one of the vertices in T.
Theorem 4.2: If a tree T admits signed product cordial labeling with n vertices then T@mK 1 (m even) is also a signed product cordial tree.
Proof: Let us assume that a tree T admits signed product cordial labeling.The mapping satisfies the condition of signed product cordial labeling.
where and .The vertex labeling for , is defined as be the m vertices joined to the vertex v of the tree T. The vertex labeling of is given by   , , , m u  , , , m vu vu vu  contributes equal number of edges labeled with -1's and 1's.Therefore in a tree T@mK 1 , the difference between the total number of vertices labeled with -1's and 1's and the difference between the total number of edges labeled with -1's and 1's differs by utmost one.
Example 4.3: Figure 2 illustrates the signed product cordial labeling for T@4K 1 where T is an arbitrary tree having signed product cordial labeling.Proof: Let ( , ) T V E  be a connected graph with vertex set V and edge set E then the tree 0 ( , Let be a vertex in T. we superimpose a pendant vertex with a selected vertex v in T only if they preserves the same sign.
As we superimpose the pendant vertex of n with any one of the vertex in T provided they preserve the same sign then the path graph contributes equal number of vertices and edges labeled with -1's and 1's.Hence the tree admits signed product cordial labeling. .For our convenience let us assume that f maps all the vertices of to 1 and all the vertices of 2 V to -1.The graph is obtained by attaching the pendant vertices to each of the vertices in G. Let the vertex set and edge set of be defined as and this labeling pattern the graph admits signed product labeling where n is a multiple of . .

Acknowledgements
The referee is gratefully acknowledged for their suggestions that improved the manuscript.
The vertex labeling for the graph , is defined as follows for, 1 } 1 i n  

References
[ All the newly added edges g u u   will share equally the labels -1 and 1 as per our construction above.Hence ( 1) (1) 0 (1) 0. n
f(v) = -1 then the sign of )

Table 2 . Vertex and edge conditions of a star graph.
then the vertex labeling for the path graph follows from theorem 2.2 1] I. Cahit, "Cordial Graphs: A Weaker Version of Graceful and Harmonious Graphs," Ars Combinatoria, Vol. 23, 1987, pp.201-207.