PA ijpam . eu ON VERTEX PRODUCT CORDIAL LABELING

In this paper, we define the vertex product cordial labeling for some path related graphs like P 2 n , path union of k copies of P 2 n graph and Pn ⊙ K1. We also discuss vertex product cordial labeling of helm graph and some graphs related to gear graph. In the present paper, we also show that the graph obtained from cycle Cn after switching of a vertex admits vertex product cordial labeling. Moreover we show that corona product Cn⊙K1, ladder graph Pn ×P2 with one edge path to one vertex, Cn ⊙ K̄m and the banana tree BT (n1, n2, n3) also admits vertex product cordial labeling with certain conditions. AMS Subject Classification: 05C78


Introduction
All graphs considered are finite, connected, undirected and simple.The graphs has the vertex set V (G) and edge set E(G) where |V (G)| = p and |E(G)| = q.Graph labeling is an assignment of integers to vertices or edges or both vertices and edges of the graph subject to certain conditions.If the domain of the mapping is the set of vertices (or edges) then the labeling is called a vertex labeling (or an edge labeling).All terms and definitions, not defined specifically in this paper, we refer to Rosa and Harary [4,5].
The origin of graph labeling is graceful labeling which was introduced by Rosa [5] in 1967.The concept of cordial labeling was introduced by Cahit [1] in 1987 as a weaker version of graceful and harmonious labeling.He write down a paper in which he investigated several results on this newly defined concept.Some labeling with variations in cordial labeling have been introduced such as prime cordial labeling, E-cordial labeling, product cordial labeling and total product cordial labeling.Cordiality behavior of numerous graph were studied by several authors [10,11,12,13].The notion of product cordial labeling was introduced by Sundaram [10] in which the absolute difference in cordial labeling is replaced by product of the vertex labels.
The present work is focused on vertex product cordial labeling of graphs.S.M. Lee and A. Liu [2] proved that all complete bipartite and all fans are cordial.Further, they proved that, the cycle C n is cordial if and only if n ≡ 2 (mod 4), the wheel W n is cordial if and only if n ≡ 3 (mod 4), n ≥ 3.In this section we provide brief summary of definitions and other required information for our investigation.
In 2004, the notion of product cordial labeling was originated by Sundaram, Ponraj and Somasundaram et al. [10].
In 2004 Sundaram, Ponraj and Somsundaram [10] proved that unicyclic graph of odd order, triangular snakes, dragons, helms and union of two paths are product cordial.They also proved that a graph with p vertices and q edges with p ≥ 4 is product cordial then q < p 2 −1 4 + 1.In 2010 Vaidya and Kanani investigated product cordial labeling for the shadow graph of cycle C n .The following type of problems can be considered in the area of vertex product cordial labeling.
• How vertex product cordial labeling is affected under various graph operations.
• Construct new families of vertex product cordial graphs by finding suitable labeling.
Vaidya, Dani, Kanani, Barasara and Vyas have investigated vertex product cordial labeling for cycle related graphs, friendship graph, middle graph of path and tensor graph.Vaidya and Barasara proved that friendship graph, cycle with one chord, cycle with twin chord are product cordial graphs [3].Vaidya and Dani [12] proved that the graph obtained by joining apex vertices of two stars is product cordial and similar results for shell and wheel graphs.The path union of k copies of cycle C n (the graph obtained by joining two copies of cycle C n by path P k ) is proved product cordial by Vaidya, Dani, Kanani [12,15].The graph obtained by joining the connected components of respective graphs by a path of arbitrary length is proved product cordial by Vaidya and Vyas [19].
Definition 1.4.A vertex of degree one is called a pendant vertex.
Definition 1.5.G + = G ⊙ K 1 is the graph obtained by joining exactly one pendant edge to every vertex of a graph G.
Definition 1.6.The corona product G 1 ⊙ G 2 of two graphs G 1 and G 2 is defined as a graph obtained by one copy of G 1 (which has p 1 vertices) and p 1 copies of G 2 and joining ith vertex of G 1 with an edge to every vertex in the ith copy of G 2 .Definition 1.7.C t n is the graph constructed from a cycle C n by joining two vertices whose distance in the cycle is t.Similarly, for the gear graph G t n is the graph constructed from the gear graph G n by joining two vertices whose distance in the gear graph is t.Definition 1.8.Let G 1 , G 2 , . . ., G n be a family of disjoint stars.Let v be a new vertex.The tree obtained by joining v to one pendant vertex of each star is called a banana tree.The class of all such trees is denoted by Definition 1.9.A vertex switching G v 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 v to every other vertex which are not adjacent to v in G.
In this paper, we prove that vertex product cordial labeling of path P 2 n , gear graph G n .We also discuss vertex product cordial labeling of gear graph G 4  n , P n ⊙ K 1 , the banana tree BT (n 1 , n 2 , n 3 ), corona product C n ⊙ K 1 , ladder graph P n × P 2 with one edge path to one vertex and C n ⊙ Km .

Main Results
With the definition of vertex product cordial graph in the previous section in hand, we are ready to study vertex product cordial labeling for certain graphs.First we discuss vertex product cordial labeling for some path related graphs.Theorem 2.1.P 2 n is vertex product cordial graph for odd n.
Proof.Let P 2 n be the graph with vertices v 1 , v 2 , . . ., v n .We have In view of the above labeling pattern, we have and e f * (1) = n − 2 and e f * (0) = n − 1 and the result holds.
Example 2.1.The graph P 2 5 and its vertex product cordial labelling is shown in Figure 1.Theorem 2.2.P 2 n is not vertex product cordial graph for even n.
Proof.The graph P 2 n has n vertices and 2n − 3 edges.Assign 1 to n/2 vertices, so that the vertex condition form the product graph is satisfied.The vertices with label 1 cause n − 3 edges to have label 1 and at most n edges to have label 0. Therefore |e f * (1) − e f * (0)| = 3.So the edge condition for vertex product cordial graph is not satisfied.
Hence P 2 n is not a vertex product cordial graph for even n.
Theorem 2.3.The path union of k copies of graph P 2 n admits vertex product cordial labeling.
Proof.Let G be the path union of k copies G 1 , G 2 , . . ., G k of P 2 n .Let {v i1 , v i2 , . . ., v in } be the vertex set and v in v i+1,1 be the edge joining G i and In view of the above labeling pattern, we have and So the vertex condition and edge conditions of vertex product cordial labeling are satisfied.
Case 2: k is odd and n is also odd.
It can be easily seen that the vertex conditions and edge conditions of product cordial labeling are satisfied either |v f (1) Remark 2.1.The path union of k copies of P 2 n does not admits vertex product cordial labeling for odd k and even n. Figure 2: Vertex product cordial labeling of path union of 2 copies of P 2 5 .
In view of the above defined labeling we have Example 2.3.Vertex product cordial labeling for P 4 ⊙ K 1 is shown in Figure 3. Proof.Let v be the apex, v 1 , v 2 , . . ., v n be the vertices of degree 4 and u 1 , u 2 , . . ., u n be the pendant vertices of H n .Then |V (H n )| = 2n + 1 and |E(H n )| = 3n.We define vertex labeling f : V (G) → {0, 1} as follows: In view of the above defined labeling pattern we have v f (1) = n + 1 and Hence, H n admits vertex product cordial labeling for even n.
Case 2: n is odd.
In view of the above labeling pattern we have v f (1) = n + 1, v f (0) = n and e f * (1) = 3n  Proof.Let u be the apex vertex of gear graph G n and v 1 , v 2 , . . ., v n and u 1 , u 2 , . . ., u n be the rim vertices.Consider the vertex labeling f : In view of the above defined labeling we have Hence the gear graph G n is vertex product cordial graph for odd n.
Example 3.2.The gear graph G 7 with its vertex cordial labelling is shown in Figure 5.

Theorem 3.3. The gear graph G 4
n with a chord admits vertex product cordial labelling for even n > 4.
Proof.Let u be the apex vertex of the gear graph G n and v 1 , v 2 , . . ., v 2n be the rim vertices of  Case 1: when n is even.

In view of the above defined labeling pattern we have
In view of the above defined labeling pattern we have . Thus the required condition of vertex product cordial labeling is satisfied.
Example 4.1.Switching of a vertex in cycle C 9 namely G 9 with its vertex product cordial graph is shown in Figure 7.

Vertex Product Cordial Labeling for some Product Related Graphs
In this section, we exhibit vertex product cordial labeling for some product related graphs like C n ⊙ K 1 , ladder graph P n × P 2 with a 1-edge path attached to its one vertex and for C n ⊙ Km .
Theorem 5.1.The corona product C n ⊙K 1 admits vertex product cordial labeling for all n.
Proof.Let G = C n ⊙ K 1 and {v 1 , v 2 , . . ., v n } be the set of cycle vertices and {u 1 , u 2 , . . ., u n } be the set of pendant vertices for the graph In view of the above defined labeling pattern we have v f   Proof.Let G be the graph obtained by attaching 1-edge path to one-vertex of a ladder graph P n × P 2 .Let V (G) = {v, v 1 , v 2 , . . .v n , u 1 , u 2 , . . ., u n } and the edge set In view of the above defined labeling In view of the above defined labeling . One can observe that in each case the labeling defined above satisfies the conditions of vertex product cordial labeling and the graph under consideration is vertex product cordial.
Illustration.For better understanding of the above defined labeling pattern the vertex product cordial labeling of ladder graph P 5 × P 2 with one chord is shown in Figure 9.In this labeling n is odd.

Proof. Let the vertex set
In view of the above defined labeling pattern we have v f (0) = mn+n   Hence the result holds for even n.Hence the Banana tree BT (n 1 , n 2 , n 3 ), n 1 = n 2 = n 3 = n is a vertex product cordial graph.

Definition 1 . 1 .
A labeling 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 .Notation.If for any 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 vertices of G having label i under f * ,
Assign 1 to nk 2 vertices, so that the vertex condition form the product cordial graph is satisfied.The vertices with label 1 cause nk − k − 2 edges to have label 1 and at most nk − k + 1 edge to have label 0. Therefore |v f * (1) − v f * (0)| = 3.So the edge condition for product cordial graph is not satisfied.Example 2.2.The product cordial labeling of path union of 2-copies of P 2 5 in Figure 2. It is the case related k even and n odd.Product cordial labeling of path union of 2 copies of P 2 5 .

Example 5 . 1 .
The Corona product C 6 ⊙ K 1 with its vertex product cordial labeling is shown in Figure8.

Figure 9 :
Figure 9: Vertex product cordial labeling of P 5 × P 2 with one chord

Case 1 :
When n is even.In view of the above defined labelingv f (1) = 3n + 4 2 = v f (0) and

Figure 10 :
Figure 10: C 6 ⊙ K 1 with its vertex product cordial labeling

Case 2 :
When n is odd.In case