Square Sum Labeling of Class of Planar Graphs Reena

A (p, q) graph G is said to be square sum, if there exists a bijection f : V (G) → {0, 1, 2, . . . , p − 1} such that the induced function f∗ : E(G) → N defined by f∗(uv) = (f(u)) + (f(v)), for every uv ∈ E(G) is injective. In this paper we proved that the planar graphs Plm,n, TBL(n, α, k, β) and higher order level joined planar grid admits square sum labeling. Also the square sum properties of several classes of graphs with many odd cycles are studied.


Introduction
Unless mentioned otherwise, by a graph we shall mean in this paper a finite, undirected, connected graph without loops or multiple edges.Terms not defined here are used in the sense of Harary [7].Labeling of a graph G is an assignment of integers either to the vertices or edges or both subject to certain conditions.A dynamic survey to know about the numerous graph labeling method is regularly updated by J.A Gallian [5].Acharya and Germina defined a square sum labeling of a (p, q)-graph G [1,2] as follows.
A connected graph with one cycle is called unicyclic.We proved that all unicyclic graphs are square sum [6].In this paper we proved that the planar graphs P l m,n , T BL(n, α, k, β) and higher order level joined planar grid admits square sum labeling.Also the square sum properties of several classes of graphs with many odd cycles are studied.
A graph is planar if it has an embedding on plane.K n , n ≤ 4 are planar.But K n for n ≥ 5 is non-planar.The complete bipartite graph K m,n is square sum if m < 4 and for all n [2].However an attempt is made to find the maximal square sum bipartite planar graph and we show that the planar graph whose definition are based on complete bipartite graphs are square sum.In [9], Babujee defines a class of planar graphs which is obtained from the complete bipartite graph K m,n , m, n ≥ 3 by removing some edges to make it planar graphs, which is called a bipartite planar class and it is denoted by P l m,n .
The graph shown in Figure 1 is a bipartite planar graph with 2m+2n−4 edges and m + n vertices.We now describe the embedding used for our proofs.Place the vertices u 1 , u 2 , . . ., u n in that order along a horizontal line segment with u 1 as the left endpoint and u n as the right endpoint as shown in Figure 1.Place the vertices v m , v m−1 , ...., v 3 , v 1 in that order along a vertical line segment with v m as the top end point and v 1 as the bottom end point so that this entire line segment is above the horizontal line segment where the vertices u 1 through u n are placed.Finally place v 2 below the horizontal line segment so that the vertices v 1 , u k , v 2 , u k+1 form a face of length 4 for 1 ≤ k ≤ n − 1.Notice that though we talk about placement along a line segment, no edges other than those mentioned in the definition are to be added.
Here we show that the planar graph P l m,n has square sum labeling.Proof.Consider the graph P l m,n (V, E), where Clearly f is injective.It is enough to check the labeling of edges with end vertices having labels (a, a+x) and (a+1, a+y) where a = f (u 1 ), x > y.If a 2 + (a + x) 2 = (a + 1) 2 + (a + y) 2 , then k 2 + 2ak + 2yk = 2a + 1, where x − y = k, has no integer solution for any positive integer value of k.Hence all the edge labels are distinct.2 Definition 1.5.Let u be a vertex of P m × P n such that deg(u) = 2. Introduce an edge between every pair of distinct vertices v, w with deg(v), deg(w where d(u, v) is the distance between u and v.The graph so obtained is defined as the level joined planar grids and it is denoted by LJ m,n .
An example of LJ 4,5 is illustrated in figure 2.
Theorem 1.6.The graph LJ m,n is square sum for all m, n ≥ 2.

Proof.
Denote the vertex at the i th row and j th column of P m × P n as v i,j .Without loss of generality we can assume m ≤ n.Construct the graph LJ m,n as illustrated in Figure 2. Let V be the vertex set of LJ m,n with p vertices.Define a function f : V → {0, 1, 2, . . ., p − 1} such that Here f is injective.With the above labeling no two of the edge labels are same as the edge labels can be arranged in an increasing order.Hence f is square sum labeling of LJ m,n for all m, n ≥ 2.

2
For m=1,n=2 we get P 1 × P 2 ≡ P 2 which is square sum.A square sum labeling of level joined planar grid LJ 4,5 using theorem 1.6 is exhibited in Figure 2. Theorem above motivate us to construct a higher order square sum graph.Let v x i,j denote the vertex v i,j in the x th copy of LJ m,n .For any integer t > 1, construct a graph by joining the vertex v x m−1,n to the vertices ,1 ; 1 ≤ x < t and denote the resulting graph as LJ t m,n (see Figure 3).The following result is the general form of Theorem 1.6.
Theorem 1.7.The graph LJ t m,n is square sum for all m, n ≥ 2 and t ≥ 1.

Proof.
Denote the vertex of the graph LJ t m,n as illustrated.Here Here f is injective.With the above defined vertex labeling the edge labels can be arranged in an increasing order.Hence f so defined is a square sum labeling of graph LJ t m,n for all m, n ≥ 2 and t ≥ 1. 2 For example, a square sum labeling of LJ t m,n when m = 4, n = 5, t = 3 using theorem 1.7 is illustrated in Figure 3.For n ≥ 2 let L n be the Cartesian product P n × P 2 of a path on n vertices with a path on two vertices.In [2] it is proved that L n is square sum for n ≥ 2. Let S = {↑, ↓} be the symbol representing the position of the block (Figure 4).Let α be a sequence of n symbols of S, i.e,α ∈ S n .We will construct a graph by tiling n-blocks side by side, with their positions indicated by α.We will denote the resulting graph by T B(α) and refer to it as a triangular belt.For simplicity we denote (↑, ↑, ↑, . . ., ↑) by ↑ n and (↓, ↓, . . ., ↓) by ↓ n .Example 1: The triangular belt corresponding to sequence α = (↑, ↓, ↓), β = (↑, ↓, ↓ , ↑), γ = (↑, ↓, ↑, ↓) and δ = (↑, ↑, ↑, ↑), respectively are shown in Figure 5.
Theorem 1.10.The graph T BL(n, α, k, β) is square sum for all α in S n and β in S k with last block is being ↑ for all k > 0.
The graph has m-cycles of length n + 1 and many 4-cycles.
Theorem 2.1.P n (+)N m is square sum for all n, m ≥ 1.
Start from v d n 2 e apply BFS algorithm and label the vertices as 0, 1, . . ., n+ m − 1 in increasing order, the order in which they are visited.It is enough to check the labeling of edges with end vertices having labels (a, a + x) and (a + 1, a + y) where a = f (v 1 ), x > y.If a 2 + (a + x) 2 = (a + 1) 2 + (a + y) 2 , then k 2 + 2ak + 2yk = 2a + 1, where x − y = k, has no integer solution for any positive integer value of k.Hence all the edge labels are distinct.
Case2.When n is even.
We will refer to J(m, n) as a Jellyfish graph.
Theorem 2.2.The Jellyfish graph J(m, n) is square sum for all m, n ≥ 0.
With the above defined labeling we can arrange the edge labels in an increasing order. 2 Example 2.3.Square sum labeling of Jellyfish graph J(3, 4) is shown in Figure 10.

Figure 1 :
Figure 1 : The class P l m,n