Odd Graceful Labeling of W -Tree WT ( n , k ) and its Disjoint Union

: Let G = ( V ( G ) , E ( G )) be a graph with p vertices and q edges. A graph G of size q is said to be odd graceful if there exists an injection λ : V ( G ) → 0 , 1 , 2 , . . . , 2 q − 1 such that assigning each edge xy the label or weight | λ ( x ) − λ ( y ) | results in the set of edge labels being 1 , 3 , 5 , . . . , 2 q − 1. This concept was introduced in 1991 by Gananajothi. In this paper, we examine the odd graceful labeling of the W -tree, denoted as WT ( n , k ).


Introduction
The graph considered in this study is finite, connected, and simple.The symbols V(G) and E(G) denote the vertex set and edge set of the graph G, respectively.Additionally, p and q represent the number of vertices and edges in G, respectively.Definition 1. [1] Rosa defined a β-valuation of a graph G with q edges as an injection from the vertices of G to the set {1, 2, . . ., q}.This valuation is such that when each edge xy is assigned the label | f (x) − f (y)|, the resulting edge labels are distinct.Definition 2. A graceful labeling f possesses the property that there exists an integer λ such that for each edge xy, either f (x) ≤ λ < f (y) or f (y) ≤ λ < f (x).
If f satisfies this condition, it is referred to as an α-labeling.
Many types of labeling exist, such as magic, total, antimagic, harmonious, graceful, and oddgraceful.This work focuses on formulating odd-graceful labeling for some new graphs.The study of graceful graphs and graceful labeling was introduced by Rosa.β-valuations are functions that produce graceful labeling.The term 'graceful labeling' became widely used after Golomb's study of such labeling several years later [3].Graceful labeling and α-labeling are among the most popular classes of graph labeling [4].It is known that not every graph is graceful.The Ringel-Kotzig conjecture [5], which posits that all trees are graceful, remains a well-known open problem.
The concept of odd graceful labeling was introduced by Gnanajothi [2].Gnanajothi proposed the conjecture that all trees are odd-graceful and provided evidence for this conjecture in trees of order up to 10. Barrientos extended this conjecture to trees of order up to 12, proving that certain graphs are odd graceful, such as: • Subdividing each edge exactly once, • Every forest whose components are caterpillars, • Every tree with a diameter of at most five, • All disjoint unions of caterpillars.
Barrientos also conjectured that every bipartite graph is odd graceful.
Eldergill generalized Gnanajothi's results on stars, demonstrating that a graph obtained by joining one endpoint from each of an odd number of paths of equal length is odd graceful [6].Gao worked on the union of graphs and proved that unions of paths, stars, cycles C m ∪ P n , C m ∪ C n , and unions of cycles of order divisible by 4 are odd or both types of graceful.
S.K. Vaidya and L.Bijukumar investigated odd graceful labeling, focusing on combining two copies of C n with a path P n and two copies of a cycle C n sharing a common edge, leading to various results.
Graph labeling has become an increasingly useful mathematical model for a wide range of applications, including ambiguities in X-Ray crystallography, communication network labeling, and circuit layout optimization.Most graph labeling problems involve three elements: 1.A set of numbers S from which the labels are chosen, 2. A rule assigning a value to each edge, 3. A condition that these values must satisfy.
Motivated by Gananajothi's research, many other graphs have been studied since then, which are odd graceful, leading to many new results.However, there still exist many interesting open problems.Inspired by the research in [2,7], we began investigating the odd graceful labeling of the W-tree and its disjoint union with other graph families.
In this work, we demonstrate that a subclass, namely the W-tree WT (n, k), admits odd graceful labeling.We also prove that the graph obtained by the disjoint union of a W-tree WT (n, k) with a path P m , a star K 1,m , a bistar B m,l , and a ladder L h are odd graceful graphs.
Firstly, we define some terms that will aid in understanding the subsequent work.

Definition 4. [4]
A path is a tree P n with vertex set V( Definition 5.
[4] Let x 0 define the central vertex of star S n , 1 ≤ n and let x i , 1 ≤ i ≤ n be its leaves.Definition 6.A bistar is a graph obtain by joining the central vertices of two star graphs.
Definition 8. Let G be a graph with the set of vertices and edges as we call it w-graph and it shall be denoted by W(n).
Definition 9. Suppose that we have k isomorphic copies of w-graphs W(n).A w-tree WT(n,k) is a tree obtain by taking a new vertex a and joining it with {d i : 1 ≤ i ≤ k}, where n ≥ 2 and k ≥ 3.

Odd-gracful labeling of w-tree
In this section, we will define the odd graceful labeling of W-tree.
Proof.Let G be a graph , we denote the vertex and edge set of G as follows: Now, we define the labeling as follows Volume 118, 51-62 3. Disjoint union of two isomorphic copies of w-tree WT (n, k) The odd graceful labeling of disjoint union of two isomorphic copies of w-tree will presented in this section.
Proof.Let G be a graph, we denote the vertex and edge set of G as follows: Now we define the labeling as follows.
Utilitas Mathematica Volume 118, 51-62 Now for the apex vertex a and a ′ , we have:

Disjoint union of W-tree WT (n, k) with other families of Graphs
The odd graceful labeling of w-tree with some other known families of graphs like path, star, bistar, lader has been formulated in this section.Proof.Let G be a graph, where q represents the total number of edges of path P m and WT (n, k) and q ′ represents the total number of edges of path p m and m represents the total number of vertices of path P m .The vertex and the edge set of G can be represented as follows: Now we define the labeling as follows.
We have different schemes for both path and w-tree, where 1 Now for the w-tree, we have the following scheme where 1 Proof.Let G ba a graph and q is the total number of edges of star K (1,m) and w-tree WT (n, k) and q ′ represents the total number of edges of star K (1,m) and m represents the total number of vertices.The vertex and edge set of graph G is follows: Volume 118, 51-62 Now we define the labeling as follows.
For star K (1,m) , we have the following scheme: where Now for the w-tree, we have the following scheme where 1 Proof.Let G ba a graph and q is the total number of edges of bistar B (m,l) and w-tree WT (n, k) and q ′ represents the total number of edges of bistar B (m,n) and m represents the total number of vertices.The vertex and edge set of graph G is follows: Volume 118, 51-62 Now we define the labeling as follows.
λ : V(G) −→ {0, 1, 2, ..., 2q − 1} Now we have different schemes for both bistar and w-tree.As we know that a bistar consist of two isomorphic copies of a star, so we discuss each star separately.Now for the first star.we have the following scheme: where Now for the second star.we have the following scheme: where Now for the w-tree, we have the following scheme where 1 ≤ i ≤ k λ(c i1 ) = 2q − q ′ − 1 − 4(i − 1)

Conclusion
In this paper, we have demonstrated that a specific subclass, the W-tree denoted by WT (n, k), admits an odd graceful labeling.Additionally, we prove that graphs obtained through the union of a W-tree WT (n, k) with various other graphs-namely, a path P m , a star K 1,m , a bistar B m,m , and a ladder

4. 1 .
Disjoint union of W-tree WT (n, k) with path P m Theorem 3. A graph G WT (n, k) ∪ P m admits the odd graceful labeling if n ≥ 2k − 2.