Cordial Labeling of Corona Product of Path Graph and Second Power of Fan Graph

A graph is said to be cordial if it has 0 - 1 labeling which satisfies particular conditions. In this paper, we construct the corona between paths and second power of fan graphs and explain the necessary and sufficient conditions for this construction to be cordial


Introduction
Labeling problem is important in graph theory.It is known that graph theory and its branches have become interesting topics for almost all fields of mathematics and also other areas of science such as chemistry, biology, physics, communication, economics, engineering, and especially computer science.A graph labeling is an assignment of integers to the vertices or edges or both.There are many contributions and different types of labeling.[1] [2] [3] [4] suppose that ( ) = is a graph, where V is the set of its vertices and E is the set of its edges.Throughout, it is assumed G is connected, finite, simple and undirected.A binary vertex labeling of G is a mapping v and 1 v be the numbers of vertices labeled by 0 and 1 in V respectively, and let 0 e and 1 e be the corresponding numbers of edge in E labeled Open Journal of Discrete Mathematics by 0 and 1 respectively.A binary vertex labeling f of G is said to be cordial if

Terminology and Notation
We introduce some notation and terminology for a graph with 4r vertices [7] [8] [9].Let r M denote the labeling 0101 01 , zero-one repeated r-times if r is even and 0101 010 if r is odd; for example, (repeated r-times) when 1 r ≥ and 1 when 0 r = .Similarly, 4 0 1 r L′ is the labe- ling 0 1100 1100 1100 1 when 1 r ≥ and 01 when 0 r = .Also, we write 0 r for the labeling 0 0 (repeated r-times) and 1 r for the labeling 1 1 (repeated r-times) [7] [8] [9] [10].For specific labeling L and M of G H where G is path and H is a second fans, we let [ ] ; L M denote the corona labeling.Ad- ditional notation that we use is the following.For a given labeling of the corona G H , we let i v and i e (for 0,1 i = ) be the numbers of labels that are i as before, we let i x and i a be the corresponding quantities for G, and we let i y and i b be those for H, which are connected to the vertices labeled 0 of G. Likewise, let i y′ and i b′ be those for H, which are connected to the vertices labeled 1 of G.In case it increases by one more vertexes, so i y′′ and i b′′ will be those for H, which are connected to the vertex labeled 1 or 0 of G.It is easy to verify that, Open Journal of Discrete Mathematics ( ) x y y x y y y y when it comes to the proof, we only need to show that, for each specified combination of labeling, 0 1 1 v v − ≤ and 0 1 1 e e − ≤ .

The Corona between Paths and Second Fans
In this section, we show that the corona between paths and second power of Fan Proof.We need to examine the following cases: Case ( ) As an example, Figure 1 illustrates  ( )       As an example, Figure 3 illustrates  ( )    ( ) Proof.We need to examine the following cases: Case (1).
( ) :11 0 ,11 0 , 00 1 , 00 1 -times , Hence, one can easily show that 0 1 0 v v − = and 0 1 1 e e − = .Thus As an example, Figure 5 illustrates    As an example, Figure 6 illustrates f u is said to be the labeling of u V ∈ .For an edge e uv E = ∈ , where , if its two vertices have the same label and ( ) * 1 f e = if they have different labels.Let us denote 0

M
′ denote the labeling 1010 10 .Sometimes, we modify the labeling r M or r M ′ by adding symbols at one end or the other (or both).We let 4r L denote the labeling 0011 0011 0011 (repeated r-times) where 1 r ≥ and, 4r L′ denote the labeling 1100 1100 1100 (repeated r-times) where 1 r ≥ and 4r S denotes the labeling 1001 1001 1001 (repeated r-times) and 4r S′ denotes the labeling 0110 0110 0110 (repeated r-times).In most cases, we then modify this by adding symbols at one end or the other (or both), thus 4 101 r L denotes the labeling 0011 0011 0011 101 (repeated r-times) when 1 r ≥ and 101 when 0 r = .Similarly, 4 1 r L′ is the labeling 1 1100 1100 1100

Figure 1 . 4 PF
Figure 1.The corona between paths and second power of Fan graphs 2 4 4 P F .

Figure 2 .
Figure 2. The corona between paths and second power of Fan graphs 4 5 2 P F .

Figure 3 . 6 2P
Figure 3.The corona between paths and second power of Fan graphs 4 6 2 P F .

Figure 4 .
Figure 4.The corona between paths and second power of Fan graphs 4 7 2 P F .

Figure 5 . 4 2P
Figure 5.The corona between paths and second power of Fan graphs 5 4 2 P F .

Figure 6 .
Figure 6.The corona between paths and second power of Fan graphs 5 5 2 P F .

Figure 7 . 6 2P
Figure 7.The corona between paths and second power of Fan graphs 5 6 2 P F .
is cordial.As an example, Figure8 illustrates

Figure 8 .
Figure 8.The corona between paths and second power of Fan graphs 5 7 2 P F .

3
need to study the following cases: Case (1).

Figure 9 . 4 2P
Figure 9.The corona between paths and second power of Fan graphs 6 4 2 P F .
[9] corona between G and H is the graph denoted by G H and is obtained by taking one copy of G and i n copies of H, and then joining the i-th vertex of G with an edge to every vertex in the i-th copy of H[9].It follows from the definition of the corona that G H has 1 1 2 Then, one can choose the labelling +. Therefore