Dividing Graceful Labeling of Certain Tree Graphs

graceful labeling of a tree defined as a simple undirected graph G(V,E) with order n and size m, if there exist an injective mapping f: V(G) → {0,1,2,3, ... , m} that induces a bijective mapping f: E(G) → {1,2,3, ... . . , m} defined by f(u, v) = |f(u) − f(v)| for each (u, v) ∈ E(G) and u , v ∈ V(G). In this paper we introduce a new type of graceful labeling denoted dividing graceful then discuss this type of certain tree graphs .


1-Introduction
A graph  = (, ) consists of two finite sets: (), the vertex set of the graph , which is nonempty set of elements called vertices, the number of these vertices called order, and () the edge set of graph (may be empty set) , which is set of elements called edge, the number of these edges called size.A graph then can be thought of as drawing or diagram consisting of collection of vertices together with edges joining certain pairs of these vertices [1].Let (, ) be a simple undirected graph with order  and size  , if there exist an injective mapping : () → {0,1,2,3, … , } that induces a bijective mapping  * : () → {1,2,3, … , } defined by  * (, ) = |() − ()| for each [, ] ∈ () and  ,  ∈ (), then  is called graceful labeling of graph  [2].In 2002, Kourosh, E. Introduced a graceful with some operation graphs [1].In 2012 Uma, R. and Murugesan, N. discussed graceful labeling of some simple graphs and their properties [3].In 2014 Pradham, P. and Kumar, K. discussed the graphs which are obtained by adding the pendant edge to the vertices of   or  2 or both in  2 +   are graceful [2].In 2014 , Munia, A. and et.al.. discussed a new class of graceful tree [4].In 2014, Vaithilingam, K. discussed difference labeling of some graph families [5].In 2018, Selvarajan, T. M. and Subramoniam, R. discussed prime graceful labeling [6].In this paper discuss a new type of graceful denoted by dividing graceful then study it for certain tree graphs.

Fig. 2 . 1
Fig. 2.1 Tree .Definition 2.4[9]: A binary tree is defined as a tree in which there is exactly one vertex of degree two and each of the remaining vertices is of degree one or three .Example: Graceful labeling of binary tree (Fig.2.2) .

Fig. 2 . 2
Fig. 2.2 Binary tree Definition 2.5 [7] : A graph is called a path if the degree () of every vertex ,  is ≤ 2 and there are no more than two end vertices.Example: Graceful labeling of path ( Fig. 2.3) .

Fig. 2 .
Fig. 2.3 path.Definition 2.6[8] : A caterpillar is a tree such that if one removes all of its leaves, the remaining graph is a path this path can be termed as back bone of the caterpillar.Example: Graceful labeling of the caterpillar (Fig.2.4) .

Fig. 2 . 4 Fig. 2 . 5
Fig. 2.4 The caterpillar Definition 2.7 [4] : In graph theory , a tree with one internal vertex and  leaves is said to be star  1, that happen to be complete bipartite graph  1 , .Example: Graceful labeling of star (Fig. 2.5)